9.15. Orthogonalsysteme


Mit der Einrichtung der Winkelmessung in 9.13 eröffnet sich ein weiteres geometrisches Konzept: die Orthogonalität, d.h. die Rechtwinkligkeit.
 

Definition:  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein euklidischer Vektorraum, v,wV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacYcacaWG3bGaeyicI4SaamOvaaaa@3AEF@ .

Wir sagen "v steht senkrecht auf w" (v ist orthogonal zu w), in Zeichen vw MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgwQiEjaadEhaaaa@3991@ , falls
 

vw=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgEHiQiaadEhacqGH9aqpcaaIWaaaaa@3A8F@ .
 

Beachte:

 

Bemerkung:  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein euklidischer Vektorraum, u,v,wV,   α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacYcacaWG2bGaaiilaiaadEhacqGHiiIZcaWGwbGaaiilaiaaysW7cqaHXoqycqGHiiIZcqWIDesOaaa@4369@ , dann gilt:
  1. vwwv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgwQiEjaadEhacaaMf8Uaeyi1HSTaaGzbVlaadEhacqGHLkIxcaWG2baaaa@42B1@
  2. 0v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgwQiEjaadAhaaaa@394F@
  3. vvv=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgwQiEjaadAhacaaMf8Uaeyi1HSTaaGzbVlaadAhacqGH9aqpcaaIWaaaaa@41C3@
  4. wv   für alle   vw=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhacaaMe8UaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaaMe8UaamODaiaaywW7cqGHuhY2caaMf8Uaam4Daiabg2da9iaaicdaaaa@4D84@
  5. wvαwv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhacaaMf8UaeyO0H4TaeqySdeMaam4DaiabgwQiEjaadAhaaaa@42C3@
  6. uv      wv(u+w)v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgwQiEjaadAhacaaMe8Uaey4jIKTaaGjbVlaadEhacqGHLkIxcaWG2bGaaGzbVlabgkDiElaaywW7caGGOaGaamyDaiabgUcaRiaadEhacaGGPaGaeyyPI4LaamODaaaa@4E55@

Beweis:

Zu 1.:  Da ∗ kommutativ ist, hat man sofort:  vw=0wv=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgEHiQiaadEhacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7caWG3bGaey4fIOIaamODaiabg2da9iaaicdaaaa@44AD@ .

Zu 2.:  0v=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgEHiQiaadAhacqGH9aqpcaaIWaaaaa@3A4D@ .

Zu 3.:  vvvv | v | 2 =0v=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgwQiEjaadAhacaaMf8Uaeyi1HSTaaGzbVlaadAhacqGHxiIkcaWG2bGaaGzbVlabgsDiBlaaywW7daabdaqaaiaadAhaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7caWG2bGaeyypa0JaaGimaaaa@5668@ .

Zu 4.:  Diese Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ " folgt direkt aus 3. und diese Richtung " MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ " steht in 2.

Zu 5.:  (αw)v=α(wv)=α0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg7aHjaadEhacaGGPaGaey4fIOIaamODaiabg2da9iabeg7aHjaacIcacaWG3bGaey4fIOIaamODaiaacMcacqGH9aqpcqaHXoqycqGHflY1caaIWaGaeyypa0JaaGimaaaa@4A14@ .

Zu 6.:  (u+w)v=uv+wv=0+0=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadwhacqGHRaWkcaWG3bGaaiykaiabgEHiQiaadAhacqGH9aqpcaWG1bGaey4fIOIaamODaiabgUcaRiaadEhacqGHxiIkcaWG2bGaeyypa0JaaGimaiabgUcaRiaaicdacqGH9aqpcaaIWaaaaa@48D2@ .
 


 

Die Orthogonalitätsrelation bietet eine neue Möglichkeit, Untervektorräume zu konstruieren. Wir betrachten zu diesem Zweck das System aller auf einer vorgegebenen Auswahl senkrecht stehenden Vektoren.
 
Definition:  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein euklidischer Vektorraum und MV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaadAfaaaa@3992@ . Die Menge
 
M ={wV|wv   für alle   vM} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaCaaaleqabaGaeyyPI4faaOGaeyypa0Jaai4EaiaadEhacqGHiiIZcaWGwbGaaiiFaiaadEhacqGHLkIxcaWG2bGaaGjbVlaabAgacaqG8dGaaeOCaiaabccacaqGHbGaaeiBaiaabYgacaqGLbGaaGjbVlaadAhacqGHiiIZcaWGnbGaaiyFaaaa@51C1@

das orthogonale Komplement (oder der Senkrechtraum) von M.
 


 
Beispiel: 
  1. V ={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaCaaaleqabaGaeyyPI4faaOGaeyypa0Jaai4EaiaaicdacaGG9baaaa@3C6C@ , denn:  wvfür alle   vw=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhacaaMc8UaaeOzaiaabYpacaqGYbGaaeiiaiaabggacaqGSbGaaeiBaiaabwgacaaMe8UaamODaiaaywW7cqGHuhY2caaMf8Uaam4Daiabg2da9iaaicdaaaa@4D82@ .
     
  2. {0} =V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaaicdacaGG9bWaaWbaaSqabeaacqGHLkIxaaGccqGH9aqpcaWGwbaaaa@3C6C@ , denn für alle wV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaadAfaaaa@3944@ gilt:  w0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaaicdaaaa@3950@ .
     
  3. In 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ ist <( 1 2 ) > =<( 2 1 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaGaeyOpa4ZaaWbaaSqabeaacqGHLkIxaaGccqGH9aqpcqGH8aapdaqadaqaauaabeqaceaaaeaacqGHsislcaaIYaaabaGaaGymaaaaaiaawIcacaGLPaaacqGH+aGpaaa@43F6@ , denn:
     
    x<( 1 2 ) > xα( 1 2 )   für alle   α x0      xα( 1 2 )   für alle   α0 x   beliebig      x( 1 2 ) x 1 +2 x 2 =0 x=( 2 x 2 x 2 )<( 2 1 )>. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B6B1@

     

    <( 2 1 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaeWaaeaafaqabeGabaaabaGaeyOeI0IaaGOmaaqaaiaaigdaaaaacaGLOaGaayzkaaGaeyOpa4daaa@3BEF@ ist dagegen nicht der Senkrechtraum zu ( 3 0 )+<( 1 2 )> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaG4maaqaaiaaicdaaaaacaGLOaGaayzkaaGaey4kaSIaeyipaWZaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaGaeyOpa4daaa@3EF1@ , denn aus x( 3 0 )+α( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgwQiEnaabmaabaqbaeqabiqaaaqaaiaaiodaaeaacaaIWaaaaaGaayjkaiaawMcaaiabgUcaRiabeg7aHnaabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaaaa@4132@ für alle α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3788@ folgt:

    (3+α) x 1 +2α x 2 =0   für alle   α 3 x 1 =0   (für   α=0)4 x 1 +2 x 2 =0   (für   α=1) x=0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@82A1@
     
    Der Senkrechtraum ist hier also der Nullraum. Zwar steht die rote Gerade senkrecht auf der grauen, aber nicht auf den Vektoren, die diese Gerade ausmachen!
     
  4. In C 0 ([0,1]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaakiaacIcacaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiaacMcaaaa@3CE0@ ist {0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFzecudaahaaWcbeqaaiabgwQiEbaakiaacUhacaaIWaGaaiyFaaaa@453D@ .
     
    Dazu zeigen wir: Keine stetige Funktion f0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgcMi5kaaicdaaaa@3955@ steht auf allen Polynomen senkrecht. Dies ergibt sich aus folgender Behauptung:
     
    Zu jedem f C 0 ([0,1]),   f0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaaicdaaaGccaGGOaGaai4waiaaicdacaGGSaGaaGymaiaac2facaGGPaGaaiilaiaaysW7caWGMbGaeyiyIKRaaGimaaaa@44F8@ , gibt es ein p MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xgHafaaa@4314@ mit fp>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgEHiQiaadchacqGH+aGpcaaIWaaaaa@3A7A@ .
     

    Beweis:
    Da f0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgcMi5kaaicdaaaa@3955@ , hat man zunächst: | f | 2 >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyOpa4JaaGimaaaa@3CAB@ . Des weiteren ist f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36D4@ als stetige Funktion auf dem geschlossenen Intervall [0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaaGymaiaac2faaaa@39CE@ beschränkt, es gibt also ein c>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg6da+iaaicdaaaa@3893@ , so dass | f(x) |c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadIhacaGGPaaacaGLhWUaayjcSdGaeyizImQaam4yaaaa@3EE9@ für alle x[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@3C4F@ .
     
    Nach dem Weierstraß'schen Approximationssatz gibt es nun zu ε= | f | 2 2c >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyypa0ZaaSaaaeaadaabdaqaaiaadAgaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaam4yaaaacqGH+aGpcaaIWaaaaa@410C@ ein Polynom p MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xgHafaaa@4314@ , so dass
     

    | f(x)p(x) |<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamiCaiaacIcacaWG4bGaaiykaaGaay5bSlaawIa7aiabgYda8iabew7aLbaa@432F@ für alle x[0,1] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaacUfacaaIWaGaaiilaiaaigdacaGGDbaaaa@3C4F@ .

    Über die Monotonie des Integrals erhält man daraus nun die folgende Abschätzung:
     
    fp =f(f(fp)) =fff(fp) = | f | 2 0 1 f(fp) | f | 2 0 1 | f(fp) | | f | 2 cε = | f | 2 2 >0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8998@

 
Bemerkung:  (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ sei ein euklidischer Vektorraum, M,NV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaacYcacaWGobGaeyOGIWSaamOvaaaa@3B15@ . Dann gilt:
  1. M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaCaaaleqabaGaeyyPI4faaaaa@3899@ ist ein Untervektorraum von V.
  2. M M {0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgMIihlaad2eadaahaaWcbeqaaiabgwQiEbaakiabgkOimlaacUhacaaIWaGaaiyFaaaa@3FC9@ .
  3. M ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3EA8@ .
  4. MN N M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaad6eacaaMf8UaeyO0H4TaaGzbVlaad6eadaahaaWcbeqaaiabgwQiEbaakiabgkOimlaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@466A@ .
  5. M =<M > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamytaiabg6da+maaCaaaleqabaGaeyyPI4faaaaa@3E65@ .

Beweis:

Zu 1.:  Es sind die drei charakterisierenden Eigenschaften nachzuprüfen:

  • 0 M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3AD7@ , denn: 0v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgwQiEjaadAhaaaa@394F@ für alle v, also erst recht für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ .
     
  • Sei u,w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacYcacaWG3bGaeyicI4SaamytamaaCaaaleqabaGaeyyPI4faaaaa@3CC3@ , d.h. uv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgwQiEjaadAhaaaa@398F@ und wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ . Folgt: (u+w)v MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadwhacqGHRaWkcaWG3bGaaiykaiabgwQiEjaadAhaaaa@3CC6@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ , d.h. u+w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabgUcaRiaadEhacqGHiiIZcaWGnbWaaWbaaSqabeaacqGHLkIxaaaaaa@3CF5@ .
     
  • Ist w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3B19@ , also: wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ , so hat man auch αwv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaam4DaiabgwQiEjaadAhaaaa@3B30@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ , d.h. αw M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3CB8@ .

Zu 2.:  Ist wM M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eacqGHPiYXcaWGnbWaaWbaaSqabeaacqGHLkIxaaaaaa@3D89@ , so hat man wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ (da w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3B19@ ), also insbesondere ww MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadEhaaaa@3992@ (da wM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eaaaa@393B@ ). Folgt: w=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daiabg2da9iaaicdaaaa@38A5@ .

Zu 3.:  Sei vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ , d.h. für jedes w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3B19@ gilt: wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ . Das bedeutet aber: w ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3E5A@ .

Zu 4.:  Ist MN MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaad6eaaaa@398A@ , so hat man für jedes w N MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad6eadaahaaWcbeqaaiabgwQiEbaaaaa@3B1A@ : wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vN MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad6eaaaa@393B@ , also erst recht : wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ . Also gilt: w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3B19@ .

Zu 5.:  Da M<M> MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlabgYda8iaad2eacqGH+aGpaaa@3B95@ , hat man nach 4. bereits: M <M > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaCaaaleqabaGaeyyPI4faaOGaey4GIKSaeyipaWJaamytaiabg6da+maaCaaaleqabaGaeyyPI4faaaaa@3F59@ . Bleibt zu zeigen: M <M > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaCaaaleqabaGaeyyPI4faaOGaeyOGIWSaeyipaWJaamytaiabg6da+maaCaaaleqabaGaeyyPI4faaaaa@3F5B@ . Sei dazu w M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaad2eadaahaaWcbeqaaiabgwQiEbaaaaa@3B19@ vorgegeben, also: wv MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgwQiEjaadAhaaaa@3991@ für alle vM MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaad2eaaaa@393A@ . Nach 5. und 6. der ersten Bemerkung steht w dann auch senkrecht auf allen Linearkombinationen von Elementen aus M. Das bedeutet: w<M > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolabgYda8iaad2eacqGH+aGpdaahaaWcbeqaaiabgwQiEbaaaaa@3D25@ .
 

Beachte:

Die Ermittlung von Senkrechträumen ist i.a. eine schwierige Aufgabe. Für Erzeugnisse im n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ steht uns jedoch ein bekanntes Verfahren zur Verfügung.

Bemerkung:  Für jede Sequenz v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ gilt:
 
< v 1 ,, v k > =Ker   ( v 1 v k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0Jaam4saiaadwgacaWGYbGaaGjbVpaabmaabaqbaeqabmqaaaqaaiaadAhadaWgaaWcbaGaaGymaaqabaaakeaacqWIUlstaeaacaWG2bWaaSbaaSqaaiaadUgaaeqaaaaaaOGaayjkaiaawMcaaaaa@4D43@ .

Beweis:

Es reicht zu zeigen:  { v 1 ,, v k } =Ker   ( v 1 v k ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccaGG9bWaaWbaaSqabeaacqGHLkIxaaGccqGH9aqpcaWGlbGaamyzaiaadkhacaaMe8+aaeWaaeaafaqabeWabaaabaGaamODamaaBaaaleaacaaIXaaabeaaaOqaaiabl6UinbqaaiaadAhadaWgaaWcbaGaam4AaaqabaaaaaGccaGLOaGaayzkaaaaaa@4D37@
x { v 1 ,, v k } x v 1             x v k x· v 1 =0            x· v k =0 ( v 1 v k )x=0 xKer   ( v 1 v k ). MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9C3F@
 

 



 
Definition:  Es sei ( (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein euklidischer Vektorraum. Eine Teilmenge OV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiabgkOimlaadAfaaaa@3994@ heißt orthonormal bzw. ein Orthonormalsystem (ON-System), falls
 
vw={ 1,   falls   v=w 0,   falls   vw für alle   v,wO MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgEHiQiaadEhacqGH9aqpdaGabaqaauaabaqaceaaaeaacaaIXaGaaiilaiaaysW7caqGMbGaaeyyaiaabYgacaqGSbGaae4CaiaaysW7caWG2bGaeyypa0Jaam4DaaqaaiaaicdacaGGSaGaaGjbVlaabAgacaqGHbGaaeiBaiaabYgacaqGZbGaaGjbVlaadAhacqGHGjsUcaWG3baaaiaaywW7caqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaaysW7caWG2bGaaiilaiaadEhacqGHiiIZcaWGpbaacaGL7baaaaa@63C5@ .
 
Ist O zusätzlich eine Basis von V, so nennt man O eine Orthonormalbasis (ON-Basis) von V.
 

Beachte:

Beispiel:  
  1. e 1 ,, e n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyzamaaBaaaleaacaWGUbaabeaaaaa@3C4F@ ist eine ON-Basis des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ .
     
  2. ( 1 2 3 2 ),( 3 2 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaWaaSWaaSqaaiaaigdaaeaacaaIYaaaaaGcbaWaaSWaaSqaamaakaaabaGaaG4maaadbeaaaSqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiqaaaqaamaalmaaleaadaGcaaqaaiaaiodaaWqabaaaleaacaaIYaaaaaGcbaGaeyOeI0YaaSWaaSqaaiaaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4180@ ist eine ON-Basis des 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ .
     
  3. ( 1 1 ),( 1 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGabaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiqaaaqaaiaaigdaaeaacaaIYaaaaaGaayjkaiaawMcaaaaa@3CB2@ ist eine Basis, aber keine ON-Basis des 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIYaaaaaaa@3842@ .
     
  4. { 1 π sin(nX)|n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EamaalaaabaGaaGymaaqaamaakaaabaGaeqiWdahaleqaaaaakiGacohacaGGPbGaaiOBaiaacIcacaWGUbGaamiwaiaacMcacaGG8bGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaOGaaiyFaaaa@46A0@ ist ein ON-System in C 0 ([0,2π]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaakiaacIcacaGGBbGaaGimaiaacYcacaaIYaGaeqiWdaNaaiyxaiaacMcaaaa@3E9E@ .
    Beim Beweis setzen wir zunächst das Additionstheorem für die Kosinusfunktion ein und erhalten für n,m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyicI4SaeSyfHu6aaWbaaSqabeaacqGHxiIkaaaaaa@3C8A@ :
     
    cos(nXmX)=cos(nX)cos(mX)+sin(nX)sin(mX) cos(nX+mX)=cos(nX)cos(mX)sin(nX)sin(mX) cos(nXmX)cos(nX+mX)=2sin(nX)sin(mX) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AAA0@

    Mit dem Integral:
    0 2π sin(nX)sin(mX) = 1 2 ( 0 2π cos((nm)X) 0 2π cos((n+m)X) ) = 1 2 { 1 nm sin((nm)X) | 0 2π 1 n+m sin((n+m)X) | 0 2π falls   nm 2π 1 2n sin(2nX) | 0 2π falls   n=m ={ 0,   falls   nm π,   falls   n=m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D698@

    hat man also: sin(nX)sin(mX)= δ nm MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacMgacaGGUbGaaiikaiaad6gacaWGybGaaiykaiabgEHiQiGacohacaGGPbGaaiOBaiaacIcacaWGTbGaamiwaiaacMcacqGH9aqpcqaH0oazdaWgaaWcbaGaamOBaiaad2gaaeqaaaaa@4795@ .

    Über dieses Ergebnis hinaus läßt sich zeigen:
    { 1 2π , 1 π sin(nX), 1 π cos(nX)|n } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EamaalaaabaGaaGymaaqaamaakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaGGSaWaaSaaaeaacaaIXaaabaWaaOaaaeaacqaHapaCaSqabaaaaOGaci4CaiaacMgacaGGUbGaaiikaiaad6gacaWGybGaaiykaiaacYcadaWcaaqaaiaaigdaaeaadaGcaaqaaiabec8aWbWcbeaaaaGcciGGJbGaai4BaiaacohacaGGOaGaamOBaiaadIfacaGGPaGaaiiFaiaad6gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaac2haaaa@5412@ ist orthonormal in C 0 ([0,2π]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaakiaacIcacaGGBbGaaGimaiaacYcacaaIYaGaeqiWdaNaaiyxaiaacMcaaaa@3E9E@ . Der Vektorraum

    < 1 2π , 1 π sin(nX), 1 π cos(nX)|n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaGaeqiWdahaleqaaaaakiaacYcadaWcaaqaaiaaigdaaeaadaGcaaqaaiabec8aWbWcbeaaaaGcciGGZbGaaiyAaiaac6gacaGGOaGaamOBaiaadIfacaGGPaGaaiilamaalaaabaGaaGymaaqaamaakaaabaGaeqiWdahaleqaaaaakiGacogacaGGVbGaai4CaiaacIcacaWGUbGaamiwaiaacMcacaGG8bGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaOGaeyOpa4daaa@541E@

    ist der Vektorraum der trigonometrischen Polynome, ein wichtiges Konzept zur Untersuchung periodischer Funktionen.

ON-Systeme bzw. ON-Basen erweisen sich als äußerst günstig, wenn es um die Errechnung von Linearkombinationen geht: Die nötigen Koeffizienten lassen sich über das Skalarprodukt bequem ermitteln. Die folgende Bemerkung beschreibt dies genauer.
 
Bemerkung:  n 1 ,, n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaaaaa@3C5E@ sei eine ON-Sequenz in V. Dann gilt für alle v< n 1 ,, n k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolabgYda8iaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@40F3@ :
 
v= α 1 n 1 ++ α k n k α i =v n i für alle   ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaam4AaaqabaGccaWGUbWaaSbaaSqaaiaadUgaaeqaaOGaaGzbVlabgsDiBlaaywW7cqaHXoqydaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWG2bGaey4fIOIaamOBamaaBaaaleaacaWGPbaabeaakiaaywW7caqGMbGaaei=aiaabkhacaqGGaGaaeyyaiaabYgacaqGSbGaaeyzaiaaysW7caWGPbGaeyizImQaam4Aaaaa@5FBC@ .

Beweis:  

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":  v n i = α 1 ( n 1 n i )++ α k ( n k n i )= α 1 ( δ 1i )++ α k ( δ ki )= α i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66D5@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":  Zunächst gibt es überhaupt eine Darstellung von v in < n 1 ,, n k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E74@ , etwa v= β 1 n 1 ++ β k n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iabek7aInaaBaaaleaacaaIXaaabeaakiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHYoGydaWgaaWcbaGaam4AaaqabaGccaWGUbWaaSbaaSqaaiaadUgaaeqaaaaa@441C@ . Nach dem gerade Bewiesenen ist dann β i =v n i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamODaiabgEHiQiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3DAB@ . Ist nun α i =v n i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamODaiabgEHiQiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3DA9@ , hat man α i = β i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeqOSdi2aaSbaaSqaaiaadMgaaeqaaaaa@3C6D@ . Also ist:
 

v= α 1 n 1 ++ α k n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iabeg7aHnaaBaaaleaacaaIXaaabeaakiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcqaHXoqydaWgaaWcbaGaam4AaaqabaGccaWGUbWaaSbaaSqaaiaadUgaaeqaaaaa@4418@ .

 

Eine erste Folgerung stellt einen Zusammenhang zur linearen Unabhängigkeit fest: Orthonormale Sequenzen sind stets linear unabhängig.
 
Bemerkung:  n 1 ,, n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaaaaa@3C5E@ sei orthonormal in V, dann gilt:
  1. n 1 ,, n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaaaaa@3C5E@ ist linear unabhängig in V.
  2. n 1 ,, n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaaaaa@3C5E@ ist eine ON-Basis von < n 1 ,, n k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@3E74@ .

Beweis: 

Zu 1.:  Ist α 1 n 1 ++ α k n k =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaamOBamaaBaaaleaacaaIXaaabeaakiabgUcaRiablAciljabgUcaRiabeg7aHnaaBaaaleaacaWGRbaabeaakiaad6gadaWgaaWcbaGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@43E1@ , so ist α i =0 n i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaiabgEHiQiaad6gadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIWaaaaa@3F32@ .

Zu 2.:  Ergibt sich direkt aus1.
 

Beachte:

n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ ist ON-Basis n i n j = δ ij MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWGUbWaaSbaaSqaaiaadMgaaeqaaOGaey4fIOIaamOBamaaBaaaleaacaWGQbaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@4533@ .

 

 
Weitere interessante Eigenschaften haben ON-Systeme in Bezug das Skalarprodukt und die damit gegebenen geometrischen Verhältnisse.

Bemerkung:  n 1 ,, n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaaaaa@3C5E@ sei eine ON-Sequenz in V, x,y< n 1 ,, n k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyicI4SaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+aaa@42A3@ mit den Koordinatenvektoren α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3788@ und β MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@378A@ . Dann gilt:
  1. xy=α·β MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgEHiQiaadMhacqGH9aqpcqaHXoqycqWIpM+zcqaHYoGyaaa@3F89@ .
  2. | x |=| α | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacqaHXoqyaiaawEa7caGLiWoaaaa@3FCF@ .
  3. (x,y)=(α,β) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOiImLaaiikaiaadIhacaGGSaGaamyEaiaacMcacqGH9aqpcqWIIiYucaGGOaGaeqySdeMaaiilaiabek7aIjaacMcaaaa@428C@   für x,y0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyiyIKRaaGimaaaa@3B15@ .

Beweis:

Zu 1.:xy =( α 1 n 1 ++ α k n k )( β 1 n 1 ++ β k n k ) = 1i,jk α i β j δ ij = i=1 k α i β i =α·β. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@83F8@

Zu 2.:  Wegen 1. hat man: | x | 2 =xx=α·α= | α | 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamiEaiabgEHiQiaadIhacqGH9aqpcqaHXoqycqWIpM+zcqaHXoqycqGH9aqpdaabdaqaaiabeg7aHbGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaaa@4C4E@ . Das ist die Behauptung.

Zu 3.:  Zunächst weiß man nach 2., dass mit x,y0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5bGaeyiyIKRaaGimaaaa@3B15@ auch α,β0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjabgcMi5kaaicdaaaa@3C5A@ ist. Mit 1. und 2. erhält man nun:
 

(x,y)= cos 1 ( xy | x || y | )= cos 1 ( α·β | α || β | )=(α,β) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FBF@ .

 

Eine besondere Rolle spielen orthonormale Sequenzen auch im Zusammenhang mit linearen Gleichungssystemen und Matrizen:
 
Bemerkung:  Es sei ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaaaa@3A6B@ eine n×m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgEna0kaad2gaaaa@39E5@ - Matrix.
  • Bilden die Spaltenvektoren von ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaaaa@3A6B@ eine orthonormale Sequenz, so ist
     
    1. Ker   ( a ·1 a ·m )={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGjbVpaabmaabaGaamyyamaaBaaaleaacqWIpM+zcaaIXaaabeaakiablAciljaadggadaWgaaWcbaGaeS4JPFMaamyBaaqabaaakiaawIcacaGLPaaacqGH9aqpcaGG7bGaaGimaiaac2haaaa@4957@ .
       
    2. Für jedes bIm   ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgIGiolGacMeacaGGTbGaaGjbVpaabmaabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaaaaa@4023@ ist das lineare Gleichungssystem ( a ij )x=b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaiaadIhacqGH9aqpcaWGIbaaaa@3D55@ eindeutig lösbar; genauer:
       
      ( a ij )x=b x j =b· a ·j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaiaadIhacqGH9aqpcaWGIbGaaGzbVlabgsDiBlaaywW7caWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamOyaiabl+y6NjaadggadaWgaaWcbaGaeS4JPFMaamOAaaqabaaaaa@4DBD@ .

  •  
  • Bilden die Zeilenvektoren von ( a ij ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaaaa@3A6B@ eine orthonormale Sequenz, so gilt für alle x< a 1· a n· > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabgYda8iaadggadaWgaaWcbaGaaGymaiabl+y6NbqabaGccqWIMaYscaWGHbWaaSbaaSqaaiaad6gacqWIpM+zaeqaaOGaeyOpa4daaa@445E@ mit Koordinatenvektor α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3788@ :
     
    1. ( a 1· a n· )x=α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaamyyamaaBaaaleaacaaIXaGaeS4JPFgabeaaaOqaaiabl6UinbqaaiaadggadaWgaaWcbaGaamOBaiabl+y6NbqabaaaaaGccaGLOaGaayzkaaGaamiEaiabg2da9iabeg7aHbaa@45D7@ .
       
    2. | ( a 1· a n· )x |=| x | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaqadaqaauaabeqadeaaaeaacaWGHbWaaSbaaSqaaiaaigdacqWIpM+zaeqaaaGcbaGaeSO7I0eabaGaamyyamaaBaaaleaacaWGUbGaeS4JPFgabeaaaaaakiaawIcacaGLPaaacaWG4baacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWG4baacaGLhWUaayjcSdaaaa@4B79@ .
       

Beweis:

Zu 1.:  Als orthonormale Sequenz ist a ·1 ,, a ·m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacqWIpM+zcaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamyyamaaBaaaleaacqWIpM+zcaWGTbaabeaaaaa@4126@ linear unabhängig. Nach einer Bemerkung in 9.4. ist daher Ker   ( a ·1 a ·m )={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadwgacaWGYbGaaGjbVpaabmaabaGaamyyamaaBaaaleaacqWIpM+zcaaIXaaabeaakiablAciljaadggadaWgaaWcbaGaeS4JPFMaamyBaaqabaaakiaawIcacaGLPaaacqGH9aqpcaGG7bGaaGimaiaac2haaaa@4957@ .

Zu 2.:  Für bIm   ( a ij )=< a ·1 ,, a ·m > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgIGiolGacMeacaGGTbGaaGjbVpaabmaabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaacqGH9aqpcqGH8aapcaWGHbWaaSbaaSqaaiabl+y6NjaaigdaaeqaaOGaaiilaiablAciljaacYcacaWGHbWaaSbaaSqaaiabl+y6Njaad2gaaeqaaOGaeyOpa4daaa@4E7C@ nutzen wir die Möglichkeit, bei orthonormalen Sequenzen die Koeffizienten einer Linearkombination durch das Skalarprodukt ausrechnen zu können:
 

( a ij )x=b x 1 a ·1 ++ x m a ·m =b x j =b· a ·j . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaaaqaamaabmaabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaacaWG4bGaeyypa0JaamOyaaqaaiabgsDiBlaaywW7aeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamyyamaaBaaaleaacqWIpM+zcaaIXaaabeaakiabgUcaRiablAciljabgUcaRiaadIhadaWgaaWcbaGaamyBaaqabaGccaWGHbWaaSbaaSqaaiabl+y6Njaad2gaaeqaaOGaeyypa0JaamOyaaqaaiabgsDiBlaaywW7aeaacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamOyaiabl+y6NjaadggadaWgaaWcbaGaeS4JPFMaamOAaaqabaGccaaMc8UaaiOlaaaaaaa@641D@
 

Zu 3.:  Man hat wieder: α i =x· a i· MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamiEaiabl+y6NjaadggadaWgaaWcbaGaamyAaiabl+y6Nbqabaaaaa@418F@ . Dies ist aber bereits die Behauptung, denn: ( a 1· a n· )x=( a 1· ·x a n· ·x ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaamyyamaaBaaaleaacaaIXaGaeS4JPFgabeaaaOqaaiabl6UinbqaaiaadggadaWgaaWcbaGaamOBaiabl+y6NbqabaaaaaGccaGLOaGaayzkaaGaamiEaiabg2da9maabmaabaqbaeqabmqaaaqaaiaadggadaWgaaWcbaGaaGymaiabl+y6NbqabaGccqWIpM+zcaWG4baabaGaeSO7I0eabaGaamyyamaaBaaaleaacaWGUbGaeS4JPFgabeaakiabl+y6NjaadIhaaaaacaGLOaGaayzkaaaaaa@575E@ .

Zu 4.:  Wegen | x |=| α | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacqaHXoqyaiaawEa7caGLiWoaaaa@3FCF@ ergibt sich 4. direkt aus 3.
 

Beachte:

 

Waren wir bisher "nur" in der Lage, gegebene Sequenzen auf Orthogonalität zu übersprüfen, so liefert der folgende Satz nun eine Methode, orthogonale Sequenzen gezielt herzustellen.
 
Satz (Gram-Schmidt Orthonormalisierungsverfahren):  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein euklidischer Vektorraum, n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3AE8@
Dann gilt:
Zu jeder linear unabhängigen Sequenz v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C71@ lässt sich eine orthonormale Sequenz n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ konstruieren, so dass
 
< n 1 ,, n k >=< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4815@ für alle kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgsMiJkaad6gaaaa@3981@ .

Dabei respektiert der Konstruktionsprozess eine bereits bestehende Orthonormalität:

Ist für ein k die Teilsequenz v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ bereits orthonormal, so gilt: n i = v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAhadaWgaaWcbaGaamyAaaqabaaaaa@3B1B@ für alle ik MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgsMiJkaadUgaaaa@397C@ .

Beweis per Induktion über n. Die Zusatzaussage wird parallel in der rechten Spalte mitbewiesen.

n=1: ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaacaaMc8UaamOBaiabg2da9iaaigdacaGG6aaaaaaa@3AF6@   Sei v 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaaaaa@37CB@ linear unabhängig; insbesondere ist v 1 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiabgcMi5kaaicdaaaa@3A56@ . Setzt man nun
 
n 1 = v 1 ° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGymaaqabaGccqGHWcaSaaa@3CAB@ ,

so ist n 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaaaaa@37C3@ eine ON-Sequenz mit < n 1 >=< v 1 > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccqGH+aGpaaa@3ED7@ .
 
        Falls | v 1 |=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG2bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdGaeyypa0JaaGymaaaa@3CB8@ , ist offenbar n 1 = v 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabg2da9iaadAhadaWgaaWcbaGaaGymaaqabaaaaa@3AB5@ .
nn+1: ¯ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaacaaMc8UaamOBaiaaywW7cqGHshI3caaMf8UaamOBaiabgUcaRiaaigdacaGG6aaaaaaa@413E@   Sei nun v 1 ,, v n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@3E0E@ eine linear unabhängige Sequenz. Also ist auch v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C71@ linear unabhängig, so dass es nach Induktionsvoraussetzung eine ON-Sequenz n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ gibt mit
 
< n 1 ,, n k >=< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGRbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH+aGpaaa@4815@ für alle kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgsMiJkaad6gaaaa@3981@ .

Wir setzen
 
n'= v n+1 ( v n+1 n 1 ) n 1 ( v n+1 n n ) n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacEcacqGH9aqpcaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTiaacIcacaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgEHiQiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGPaGaeyyXICTaamOBamaaBaaaleaacaaIXaaabeaakiabgkHiTiablAciljabgkHiTiaacIcacaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgEHiQiaad6gadaWgaaWcbaGaamOBaaqabaGccaGGPaGaeyyXICTaamOBamaaBaaaleaacaWGUbaabeaaaaa@58D3@

und berechnen für jn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgsMiJkaad6gaaaa@3980@ :
 
n' n j = v n+1 n j ( v n+1 n 1 )( n 1 n j )( v n+1 n n )( n n n j ) = v n+1 n j ( v n+1 n j )( n j n j ) = v n+1 n j v n+1 n j =0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9353@

Weil nun v n+1 < v 1 ,, v n >=< n 1 ,, n n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHjiYZcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4Jaeyypa0JaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaakiabg6da+aaa@4D62@ , hat man: n'0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacEcacqGHGjsUcaaIWaaaaa@3A08@ . Setzt man jetzt
 

n n+1 =n'° MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGUbGaai4jaiabgclaWcaa@3E32@ ,

so ist zunächst n 1 ,, n n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@3DFE@ eine ON-Sequenz.

Die Information n n+1 v n+1 < n 1 ,, n n >=< v 1 ,, v n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgIGiolabgYda8iaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcqGH9aqpcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4daaa@5206@ deuten wir zweifach:

n n+1 < v 1 ,, v n , v n+1 > v n+1 < n 1 ,, n n , n n+1 > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiaad6gadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyicI4SaeyipaWJaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabg6da+aqaaiaadAhadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyicI4SaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaakiaacYcacaWGUbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabg6da+aaaaaa@5A7D@

und leiten daraus ab:
 

< n 1 ,, n n , n n+1 >< v 1 ,, v n , v n+1 >< n 1 ,, n n , n n+1 > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@60E8@ .

Also gilt mit der Induktionsvoraussetzung:
 
< n 1 ,, n k >=< v 1 ,, v k > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabeqaaaqaaiabgYda8iaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaam4AaaqabaGccqGH+aGpcqGH9aqpcqGH8aapcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaeyOpa4daaaaa@481F@ für alle kn+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgsMiJkaad6gacqGHRaWkcaaIXaaaaa@3B1E@ .

 

  Sei jetzt eine Teilsequenz v 1 ,, v k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGRbaabeaaaaa@3C6E@ von v 1 ,, v n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@3E0E@ bereits orthonormal. Falls kn MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgsMiJkaad6gaaaa@3981@ , ist nichts zu zeigen. Es reicht daher, nur den Fall k=n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaad6gacqGHRaWkcaaIXaaaaa@3A6F@ zu betrachten: v 1 ,, v n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@3E0E@ ist bereits orthonormal.

Damit ist auch v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C71@ eine ON-Sequenz. Nach Induktionsvoraussetzung gilt daher:
 

n i = v i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAhadaWgaaWcbaGaamyAaaqabaaaaa@3B1B@ für alle in MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgsMiJkaad6gaaaa@397F@ .

Also hat man:
 
n'= v n+1 i=1 n ( v n+1 v i ) =0 v i = v n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacEcacqGH9aqpcaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTmaaqahabaWaaGbaaeaacaGGOaGaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHxiIkcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaWcbaGaeyypa0JaaGimaaGccaGL44pacqGHflY1caWG2bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyypa0JaamODamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@5822@

und damit: n n+1 = v n+1 °= v n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWG2bWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgclaWkabg2da9iaadAhadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@4512@ .

 
Beispiel:  Wir orthonormalisieren in 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa@3843@ die linear unabhängige Sequenz ( 0 0 2 ),( 1 1 1 ),( 1 3 2 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeWabaaabaGaaGimaaqaaiaaicdaaeaacaaIYaaaaaGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqadeaaaeaacaaIXaaabaGaaGymaaqaaiaaigdaaaaacaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIZaaabaGaaGOmaaaaaiaawIcacaGLPaaaaaa@42A6@ . Dabei benutzen wir die Konstruktionen aus dem Induktionsbeweis und errechen die Vektoren n1,n2,n3 der Reihe nach:

1.:  n 1 = ( 0 0 2 ) ° = 1 2 ( 0 0 2 )=( 0 0 1 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabg2da9maabmaabaqbaeqabmqaaaqaaiaaicdaaeaacaaIWaaabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabgclaWcaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaafaqabeWabaaabaGaaGimaaqaaiaaicdaaeaacaaIYaaaaaGaayjkaiaawMcaaiabg2da9maabmaabaqbaeqabmqaaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@49E0@ .

2.:  n'=( 1 1 1 )(( 1 1 1 )· n 1 ) n 1 =( 1 1 1 )1( 0 0 1 )=( 1 1 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@57A9@ , also:  n 2 = ( 1 1 0 ) ° =( 1 2 1 2 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIYaaabeaakiabg2da9maabmaabaqbaeqabmqaaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabgclaWcaakiabg2da9maabmaabaqbaeqabmqaaaqaamaalmaaleaacaaIXaaabaWaaOaaaeaacaaIYaaameqaaaaaaOqaamaalmaaleaacaaIXaaabaWaaOaaaeaacaaIYaaameqaaaaaaOqaaiaaicdaaaaacaGLOaGaayzkaaaaaa@458D@ .

3.:  n'=( 1 3 2 )(( 1 3 2 )· n 1 ) n 1 (( 1 3 2 )· n 2 ) n 2 =( 1 3 2 )2( 0 0 1 ) 4 2 ( 1 2 1 2 0 )=( 1 1 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6D50@ , also  n 3 = ( 1 1 0 ) ° =( 1 2 1 2 0 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIZaaabeaakiabg2da9maabmaabaqbaeqabmqaaaqaaiabgkHiTiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabgclaWcaakiabg2da9maabmaabaqbaeqabmqaaaqaaiabgkHiTmaalmaaleaacaaIXaaabaWaaOaaaeaacaaIYaaameqaaaaaaOqaamaalmaaleaacaaIXaaabaWaaOaaaeaacaaIYaaameqaaaaaaOqaaiaaicdaaaaacaGLOaGaayzkaaaaaa@4768@ .

 
Beispiel:  In C 0 ([0,1]) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGimaaaakiaacIcacaGGBbGaaGimaiaacYcacaaIXaGaaiyxaiaacMcaaaa@3CE0@ orthonormalisieren wir die linear unabhängige Sequenz 1,X, X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacYcacaWGybGaaiilaiaadIfadaahaaWcbeqaaiaaikdaaaaaaa@3AA7@ :

1.:  | 1 | 2 = 0 1 1 2 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaaIXaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Zaa8qCaeaacaaIXaWaaWbaaSqabeaacaaIYaaaaaqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccqGH9aqpcaaIXaaaaa@4301@ , also  n 1 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiabg2da9iaaigdaaaa@398E@ .

2.:  Mit X1= 0 1 X1 = 1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgEHiQiaaigdacqGH9aqpdaWdXbqaaiaadIfacqGHflY1caaIXaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@43CD@ errechnet man zunächst: n'=X(X1)1=X 1 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacEcacqGH9aqpcaWGybGaeyOeI0IaaiikaiaadIfacqGHxiIkcaaIXaGaaiykaiaaigdacqGH9aqpcaWGybGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@4349@ .
Das Längenquadrat | n' | 2 = 0 1 (X 1 2 ) 2 = 1 12 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGUbGaai4jaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaakiabg2da9maapehabaGaaiikaiaadIfacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacMcadaahaaWcbeqaaiaaikdaaaaabaGaaGimaaqaaiaaigdaa0Gaey4kIipakiabg2da9maalaaabaGaaGymaaqaaiaaigdacaaIYaaaaaaa@495A@ ergibt schließlich:

n 2 = 12 (X 1 2 )=2 3 X 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIYaaabeaakiabg2da9maakaaabaGaaGymaiaaikdaaSqabaGccaaMc8UaaiikaiaadIfacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiaacMcacqGH9aqpcaaIYaWaaOaaaeaacaaIZaaaleqaaOGaamiwaiabgkHiTmaakaaabaGaaG4maaWcbeaaaaa@45EB@ .

3.:  Über X 2 1= 0 1 X 2 1 = 1 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgEHiQiaaigdacqGH9aqpdaWdXbqaaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHflY1caaIXaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaG4maaaaaaa@45B4@   und  X 2 (2 3 X 3 )=2 3 0 1 X 3 3 0 1 X 2 = 3 6 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgEHiQiaacIcacaaIYaWaaOaaaeaacaaIZaaaleqaaOGaamiwaiabgkHiTmaakaaabaGaaG4maaWcbeaakiaacMcacqGH9aqpcaaIYaWaaOaaaeaacaaIZaaaleqaaOWaa8qCaeaacaWGybWaaWbaaSqabeaacaaIZaaaaaqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccqGHsisldaGcaaqaaiaaiodaaSqabaGcdaWdXbqaaiaadIfadaahaaWcbeqaaiaaikdaaaaabaGaaGimaaqaaiaaigdaa0Gaey4kIipakiabg2da9maalaaabaWaaOaaaeaacaaIZaaaleqaaaGcbaGaaGOnaaaaaaa@50BD@ ergibt sich zunächst wieder:

n'= X 2 ( X 2 1)1( X 2 (2 3 X 3 ))(2 3 X 3 )= X 2 1 3 X+ 1 2 = X 2 X+ 1 6 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@61AE@ .

Wir errechnen noch einmal ein Längenquadrat: | n' | 2 = 0 1 ( X 2 X+ 1 6 ) = 1 180 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGUbGaai4jaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaakiabg2da9maapehabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGybGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOnaaaacaGGPaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGymaiaaiIdacaaIWaaaaaaa@4BF2@ , und erhalten daraus:
n 3 = 180 ( X 2 X+ 1 6 )=6 5 X 2 6 5 X+ 5 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIZaaabeaakiabg2da9maakaaabaGaaGymaiaaiIdacaaIWaaaleqaaOGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGybGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOnaaaacaGGPaGaeyypa0JaaGOnamaakaaabaGaaGynaaWcbeaakiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI2aWaaOaaaeaacaaI1aaaleqaaOGaamiwaiabgUcaRmaakaaabaGaaGynaaWcbeaaaaa@4C35@ .

 

Für endliche, euklidische Vektorräume ergeben sich aus dem Gram-Schmidt Verfahren wichtige Folgerungen:
 
Bemerkung:  
  1. Jeder endliche, euklidische Vektorraum V besitzt eine ON-Basis.
     
  2. In einem endlichen, euklidischen Vektorraum V lässt sich jede orthonormale Sequenz n 1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@3C60@ zu einer ON-Basis ergänzen.

Beweis:

Zu 1.:  Zunächst besitzt V - wie jeder Vektorraum - eine Basis, etwa v 1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@3C71@ . Die orthonormalisierte Sequenz n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ ist dann eine ON-Basis, denn < n 1 ,, n n >=< v 1 ,, v n >=V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaakiabg6da+iabg2da9iabgYda8iaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaadAhadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcqGH9aqpcaWGwbaaaa@49FC@ .

Zu 2.:  Als orthonormale Sequenz ist n 1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@3C60@ insbesondere linear unabhängig, lässt sich also nach Basisergänzungssatz zu einer Basis n 1 ,, n m , v m+1 ,, v n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiaacYcacaWG2bWaaSbaaSqaaiaad2gacqGHRaWkcaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamODamaaBaaaleaacaWGUbaabeaaaaa@4576@ ergänzen und anschließend, wie in 1., in eine ON-Basis umschreiben. Dabei garantiert das Orthonormalisierungverfahren, dass die Vektoren n 1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@3C60@ unverändert bleiben.
 

 
Bemerkung:  Es sei n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ eine ON-Basis von V, dann gilt für jedes m{0,,n} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgIGiolaacUhacaaIWaGaaiilaiablAciljaacYcacaWGUbGaaiyFaaaa@3E8E@ :
 
< n 1 ,, n m > =< n m+1 ,, n n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaWGTbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamOBaaqabaGccqGH+aGpaaa@4BC6@ .

Beweis:

O.E. 0<m<n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgYda8iaad2gacqGH8aapcaWGUbaaaa@3A90@ . Ferner reicht es zu zeigen: { n 1 ,, n m } =< n m+1 ,, n n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaad6gadaWgaaWcbaGaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamyBaaqabaGccaGG9bWaaWbaaSqabeaacqGHLkIxaaGccqGH9aqpcqGH8aapcaWGUbWaaSbaaSqaaiaad2gacqGHRaWkcaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaakiabg6da+aaa@4BBA@ .

Da n i n j =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGPbaabeaakiabgEHiQiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@3CC7@ für im<j MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgsMiJkaad2gacqGH8aapcaWGQbaaaa@3B71@ , hat man zunächst: n m+1 ,, n n { n 1 ,, n m } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaWGTbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamOBaaqabaGccqGHiiIZcaGG7bGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiaac2hadaahaaWcbeqaaiabgwQiEbaaaaa@4A22@ und damit: < n m+1 ,, n n > { n 1 ,, n m } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamOBamaaBaaaleaacaWGTbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamOBaaqabaGccqGH+aGpcqGHckcZcaGG7bGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiaac2hadaahaaWcbeqaaiabgwQiEbaaaaa@4CA6@ .

Sei nun v { n 1 ,, n m } MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaacUhacaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacaWGUbWaaSbaaSqaaiaad2gaaeqaaOGaaiyFamaaCaaaleqabaGaeyyPI4faaaaa@42C7@ , also v n i =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgEHiQiaad6gadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIWaaaaa@3BAA@ für im MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgsMiJkaad2gaaaa@397E@ . Da n 1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@3C61@ ist eine ON-Basis ist, hat man für v die folgende Darstellung:

v =(v n 1 ) n 1 ++(v n m ) n m +(v n m+1 ) n m+1 ++(v n n ) n n =(v n m+1 ) n m+1 ++(v n n ) n n < n m+1 ,, n n >. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@820E@

 

 
Bemerkung:  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein endlicher, euklidischer Vektorraum, WV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabgkOimlaadAfaaaa@399C@ ein Untervektorraum. Dann gilt:
  1. W W ={0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabgMIihlaadEfadaahaaWcbeqaaiabgwQiEbaakiabg2da9iaacUhacaaIWaGaaiyFaaaa@3EE7@ .
  2. Zu jedem vV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaadAfaaaa@3943@ gibt es genau zwei Vektoren v'W,   v" W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacEcacqGHiiIZcaWGxbGaaiilaiaaysW7caWG2bGaaiOiaiabgIGiolaadEfadaahaaWcbeqaaiabgwQiEbaaaaa@420B@ , so dass v=v'+v" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaadAhacaGGNaGaey4kaSIaamODaiaackcaaaa@3C13@ .

Beweis:

Zu 1.:  Man weiß bereits: W W {0} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabgMIihlaadEfadaahaaWcbeqaaiabgwQiEbaakiabgkOimlaacUhacaaIWaGaaiyFaaaa@3FDD@ . Da aber W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36C5@ und W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaCaaaleqabaGaeyyPI4faaaaa@38A3@ Untervektorräume sind, enthalten sie den Nullvektor. Also gilt auch: {0}W W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaaicdacaGG9bGaeyOGIWSaam4vaiabgMIihlaadEfadaahaaWcbeqaaiabgwQiEbaaaaa@3FD3@ .

Zu 2.:  Mit V ist auch W ein endlicher, euklidischer Vektorraum. W besitzt somit eine ON-Basis n 1 ,, n m MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaaaaa@3C60@ , die wir zu einer ON-Basis n 1 ,, n m , n m+1 ,, n n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiaacYcacaWGUbWaaSbaaSqaaiaad2gacqGHRaWkcaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGUbaabeaaaaa@4566@ von V ergänzen. Nach der Bemerkung zuvor ist
 

W =< n 1 ,, n m > =< n m+1 ,, n n > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaaIXaaabeaakiaacYcacqWIMaYscaGGSaGaamOBamaaBaaaleaacaWGTbaabeaakiabg6da+maaCaaaleqabaGaeyyPI4faaOGaeyypa0JaeyipaWJaamOBamaaBaaaleaacaWGTbGaey4kaSIaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiaad6gadaWgaaWcbaGaamOBaaqabaGccqGH+aGpaaa@4F90@ ,
so dass die Vektoren
 
v'=(v n 1 ) n 1 ++(v n m ) n m W v"=(v n m+1 ) n m+1 ++(v n n ) n n W MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6864@

eine eindeutige Zerlegung von v darstellen: v=v'+v" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaadAhacaGGNaGaey4kaSIaamODaiaackcaaaa@3C13@ .
 

Beachte:

Ergänzend zur Aussage M ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3EA8@ lässt sich nun für endliche Vektorräume genau beschreiben, in welchen Fällen diese Teilmengenbeziehung zu einer Gleichheit erweitert werden kann:
 
Bemerkung:  Es sei (V,+,,) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAfacaGGSaGaey4kaSIaaiilaiabgwSixlaacYcacqGHxiIkcaGGPaaaaa@3E48@ ein endlicher, euklidischer Vektorraum, dann gilt:
 
M= ( M ) M    MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaGccaaMf8Uaeyi1HSTaaGzbVlaad2eacaaMe8oaaa@4593@ ist ein Untervektorraum.

Beweis:

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@3846@ ":  Hier ist nichts zu zeigen, denn ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2eadaahaaWcbeqaaiabgwQiEbaakiaacMcadaahaaWcbeqaaiabgwQiEbaaaaa@3BDA@ ist als orthogonales Komplement ein Untervektorraum.

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@3842@ ":  Da die Inklusion M ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgkOimlaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3EA8@ stets erfüllt ist, reicht es zu zeigen: M ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabgoOijlaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3EA6@ . Sei dazu v ( M ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaacIcacaWGnbWaaWbaaSqabeaacqGHLkIxaaGccaGGPaWaaWbaaSqabeaacqGHLkIxaaaaaa@3E59@ vorgegeben. Ist nun M ein Untervektorraum, so ist V=M M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiabg2da9iaad2eacqGHvksXcaWGnbWaaWbaaSqabeaacqGHLkIxaaaaaa@3D54@ . Es gibt also eine Zerlegung v=v'+v" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaadAhacaGGNaGaey4kaSIaamODaiaackcaaaa@3C13@ mit v'M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaacEcacqGHiiIZcaWGnbaaaa@39E5@ , v" M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaackcacqGHiiIZcaWGnbWaaWbaaSqabeaacqGHLkIxaaaaaa@3BBE@ . Man hat also insbesondere: vv" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgwQiEjaadAhacaGGIaaaaa@3A36@ und v"v' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaackcacqGHLkIxcaWG2bGaai4jaaaa@3AE1@ . Daher gilt:
 

0=vv"=v'v"+v"v"=v"v" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaadAhacqGHxiIkcaWG2bGaaiOiaiabg2da9iaadAhacaGGNaGaey4fIOIaamODaiaackcacqGHRaWkcaWG2bGaaiOiaiabgEHiQiaadAhacaGGIaGaeyypa0JaamODaiaackcacqGHxiIkcaWG2bGaaiOiaaaa@4ABA@ .

Also ist v"=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiaackcacqGH9aqpcaaIWaaaaa@394A@ und somit v=v'M MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaadAhacaGGNaGaeyicI4Saamytaaaa@3BE6@ .
 

 


 9.14
9.16.