9.14. Der Rieszsche Darstellungssatz


 

Das Skalarprodukt eines euklidischen Vektorraums ist in jeder Koordinate linear; dies lässt sich auch so beschreiben: Für jedes wV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaadAfaaaa@3944@ ist die Funktion
 

S w :V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaakiaacQdacaWGwbGaeyOKH4QaeSyhHekaaa@3CE9@ , gegeben durch S w (x)=wx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaakiaacIcacaWG4bGaaiykaiabg2da9iaadEhacqGHxiIkcaWG4baaaa@3E37@

eine lineare Abbildung von V nach MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ , also eine Linearform: S w V' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaakiabgIGiolaadAfacaGGNaaaaa@3AFD@

In diesem Abschnitt werden wir zeigen, dass den Linearformen des Typs S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaaaaa@37E9@ eine besondere Bedeutung zukommt: 

Wir beginnen mit einer Eindeutigkeitsüberlegung.

Bemerkung:  Es sei V ein euklidischer Vektorraum. Die Abbildung
 
w S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiablAAiHjaadofadaWgaaWcbaGaam4Daaqabaaaaa@3A9E@

ist injektiv, d.h. S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaaaaa@37E9@ ist durch seinen Vektor w eindeutig bestimmt.

Beweis:

Wäre nämlich S w = S w' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaakiabg2da9iaadofadaWgaaWcbaGaam4DaiaacEcaaeqaaaaa@3BA4@ mit einem weiteren w'V MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacEcacqGHiiIZcaWGwbaaaa@39EF@ , so hätte man für alle xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3945@ :
 

wx=w'x (ww')x=0. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaaqaaaqaaiaadEhacqGHxiIkcaWG4bGaeyypa0Jaam4DaiaacEcacqGHxiIkcaWG4baabaGaeyi1HSTaaGzbVdqaaiaacIcacaWG3bGaeyOeI0Iaam4DaiaacEcacaGGPaGaey4fIOIaamiEaiabg2da9iaaicdacaGGUaaaaaaa@4AAA@
 
Also ist insbesondere:
 
(ww')(ww')=0 | ww' | 2 =0 ww'=0 w=w' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaaaabaaabaGaaiikaiaadEhacqGHsislcaWG3bGaai4jaiaacMcacqGHxiIkcaGGOaGaam4DaiabgkHiTiaadEhacaGGNaGaaiykaiabg2da9iaaicdaaeaacqGHuhY2caaMf8oabaWaaqWaaeaacaWG3bGaeyOeI0Iaam4DaiaacEcaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIWaaabaGaeyi1HSTaaGzbVdqaaiaadEhacqGHsislcaWG3bGaai4jaiabg2da9iaaicdaaeaacqGHuhY2caaMf8oabaGaam4Daiabg2da9iaadEhacaGGNaaaaaaa@609B@

 


Nun zur ersten Aussage über die Linearformen des n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacaWGUbaaaaaa@3879@ :
 
Bemerkung:  Zu jeder Linearform f n ' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaakiaacEcaaaa@3B9D@ gibt es genau ein w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AF9@ , so dass f= S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadofadaWgaaWcbaGaam4Daaqabaaaaa@39DA@ .

Beweis: Die Eindeutigkeit ist bereits in der Vorüberlegung nachgewiesen. Es ist also nur noch ein geeignetes w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AF9@ zu konstruieren. Setzt man nun
 

w i =f( e i ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAgacaGGOaGaamyzamaaBaaaleaacaWGPbaabeaakiaacMcaaaa@3D61@ ,

so hat man für jedes x n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolabl2riHoaaCaaaleqabaGaamOBaaaaaaa@3AFA@ :

f(x)= i=1 n x i f( e i ) = i=1 n x i w i =x·w= S w (x) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9maaqahabaGaamiEamaaBaaaleaacaWGPbaabeaakiaadAgacaGGOaGaamyzamaaBaaaleaacaWGPbaabeaakiaacMcaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaeyypa0ZaaabCaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaam4DamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabg2da9iaadIhacqWIpM+zcaWG3bGaeyypa0Jaam4uamaaBaaaleaacaWG3baabeaakiaacIcacaWG4bGaaiykaaaa@5C86@ .
 

 
Wie bereits angedeutet ist eine Übertragung dieses Ergebnisses auf beliebige euklidische Vektorräume nur eingeschränkt möglich: Die Vektorräume müssen eine bestimmte "topologische" Qualität besitzen und auch nicht jede Linearform lässt sich in der Form S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWG3baabeaaaaa@37E9@ notieren. Die genauen Verhältnisse beschreibt der Rieszsche Darstellungssatz.

Wir benötigen zunächst eine technische Vorübereitung:
 
Bemerkung:  Es sei V ein euklidischer Vektorraum,  x und w zwei feste Vektoren aus V, w0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgcMi5kaaicdaaaa@3966@ . Dann sind die Abbildungen φ,ψ: MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdyMaaiilaiabeI8a5jaacQdacqWIDesOcqGHsgIRcqWIDesOaaa@3FAB@ , gegeben durch
 
φ(α)=| wαx | ψ(α)=| w+αx | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiqaaaqaaiabeA8aMjaacIcacqaHXoqycaGGPaGaeyypa0ZaaqWaaeaacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aaqaaiabeI8a5jaacIcacqaHXoqycaGGPaGaeyypa0ZaaqWaaeaacaWG3bGaey4kaSIaeqySdeMaamiEaaGaay5bSlaawIa7aaaaaaa@50BB@

differenzierbar in 0. Dabei ist φ'(0)=w°x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdyMaai4jaiaacIcacaaIWaGaaiykaiabg2da9iabgkHiTiaadEhacqGHWcaScqGHxiIkcaWG4baaaa@4127@ und ψ'(0)=w°x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdKNaai4jaiaacIcacaaIWaGaaiykaiabg2da9iaadEhacqGHWcaScqGHxiIkcaWG4baaaa@404F@ .

Beweis:

Wir führen den Beweis (nur) für die Funktion φ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdygaaa@37A2@ und setzen dazu die Differenzenquotientenfunktion an:

φ(α)φ(0) α = | wαx || w | α = (| wαx || w |)(| wαx |+| w |) α(| wαx |+| w |) = | wαx | 2 | w | 2 α(| wαx |+| w |) = | w | 2 2αwx+ α 2 | x | 2 | w | 2 α(| wαx |+| w |) = 2wx+α | x | 2 | wαx |+| w | . MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuGaaaaabaWaaSaaaeaacqaHgpGzcaGGOaGaeqySdeMaaiykaiabgkHiTiabeA8aMjaacIcacaaIWaGaaiykaaqaaiabeg7aHbaaaeaacqGH9aqpdaWcaaqaamaaemaabaGaam4DaiabgkHiTiabeg7aHjaadIhaaiaawEa7caGLiWoacqGHsisldaabdaqaaiaadEhaaiaawEa7caGLiWoaaeaacqaHXoqyaaaabaaabaGaeyypa0ZaaSaaaeaacaGGOaWaaqWaaeaacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aiabgkHiTmaaemaabaGaam4DaaGaay5bSlaawIa7aiaacMcacaGGOaWaaqWaaeaacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aiabgUcaRmaaemaabaGaam4DaaGaay5bSlaawIa7aiaacMcaaeaacqaHXoqycaGGOaWaaqWaaeaacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aiabgUcaRmaaemaabaGaam4DaaGaay5bSlaawIa7aiaacMcaaaaabaaabaGaeyypa0ZaaSaaaeaadaabdaqaaiaadEhacqGHsislcqaHXoqycaWG4baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaqWaaeaacaWG3baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaGcbaGaeqySdeMaaiikamaaemaabaGaam4DaiabgkHiTiabeg7aHjaadIhaaiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaadEhaaiaawEa7caGLiWoacaGGPaaaaaqaaaqaaiabg2da9maalaaabaWaaqWaaeaacaWG3baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGOmaiabeg7aHjaadEhacqGHxiIkcaWG4bGaey4kaSIaeqySde2aaWbaaSqabeaacaaIYaaaaOWaaqWaaeaacaWG4baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaqWaaeaacaWG3baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaGcbaGaeqySdeMaaiikamaaemaabaGaam4DaiabgkHiTiabeg7aHjaadIhaaiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaadEhaaiaawEa7caGLiWoacaGGPaaaaaqaaaqaaiabg2da9maalaaabaGaeyOeI0IaaGOmaiaadEhacqGHxiIkcaWG4bGaey4kaSIaeqySde2aaqWaaeaacaWG4baacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaGcbaWaaqWaaeaacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aiabgUcaRmaaemaabaGaam4DaaGaay5bSlaawIa7aaaacaGGUaaaaaaa@E19B@

Dieser Quotient besitzt einen Grenzwert für α0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOKH4QaaGimaaaa@3A2F@ , und zwar:

φ'(0)= 2wx | w |+| w | = w | w | x=w°x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdyMaai4jaiaacIcacaaIWaGaaiykaiabg2da9maalaaabaGaeyOeI0IaaGOmaiaadEhacqGHxiIkcaWG4baabaWaaqWaaeaacaWG3baacaGLhWUaayjcSdGaey4kaSYaaqWaaeaacaWG3baacaGLhWUaayjcSdaaaiabg2da9iabgkHiTmaalaaabaGaam4DaaqaamaaemaabaGaam4DaaGaay5bSlaawIa7aaaacqGHxiIkcaWG4bGaeyypa0JaeyOeI0Iaam4DaiabgclaWkabgEHiQiaadIhaaaa@58F5@ .
 


 
Satz (Rieszscher Darstellungssatz):  Es sei V ein vollständiger euklidischer Vektorraum. Ist fV' MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadAfacaGGNaaaaa@39DE@ eine beschränkte Linearform, d.h. gibt es ein c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiolabl2riHcaa@39C5@ , so dass
 
| f(v) |c| v | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadAhacaGGPaaacaGLhWUaayjcSdGaeyizImQaam4yamaaemaabaGaamODaaGaay5bSlaawIa7aaaa@4304@   für alle vV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgIGiolaadAfaaaa@3943@ ,

so gibt es genau ein wV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaadAfaaaa@3944@ , so dass f= S w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadofadaWgaaWcbaGaam4Daaqabaaaaa@39DA@ .

Beweis:

Da die Eindeutigkeit bereits gegeben ist, reicht es wieder, ein geeignetes w zu konstruieren. Wir betrachten dazu zunächst die Menge

M={| f(x) ||| x |=1} MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2da9iaacUhadaabdaqaaiaadAgacaGGOaGaamiEaiaacMcaaiaawEa7caGLiWoacaGG8bWaaqWaaeaacaWG4baacaGLhWUaayjcSdGaeyypa0JaaGymaiaac2hacqGHckcZcqWIDesOaaa@4A70@ .

Nach Voraussetzung ist dies eine nicht-leere Teilmenge von [0,c] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaicdacaGGSaGaam4yaiaac2faaaa@39FB@ . M besitzt also gemäß Vollständigkeitsaxiom ein Supremum:
 

s=supM[0,c] MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2da9iGacohacaGG1bGaaiiCaiaad2eacqGHiiIZcaGGBbGaaGimaiaacYcacaWGJbGaaiyxaaaa@4135@ .

Die vorgegebene Abschätzung | f(v) |c| v | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadAhacaGGPaaacaGLhWUaayjcSdGaeyizImQaam4yamaaemaabaGaamODaaGaay5bSlaawIa7aaaa@4304@ lässt sich - wegen | f(v) |=| f(| v |v°) |=| v || f(v°) | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadAhacaGGPaaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWGMbGaaiikamaaemaabaGaamODaaGaay5bSlaawIa7aiaadAhacqGHWcaScaGGPaaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWG2baacaGLhWUaayjcSdWaaqWaaeaacaWGMbGaaiikaiaadAhacqGHWcaScaGGPaaacaGLhWUaayjcSdaaaa@572A@ - verschärfen zu:
 

| f(v) |s| v | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadAhacaGGPaaacaGLhWUaayjcSdGaeyizImQaam4CamaaemaabaGaamODaaGaay5bSlaawIa7aaaa@4314@ .

Falls nun s=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2da9iaaicdaaaa@38A1@ , ist auch f=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaaicdaaaa@3894@ . In diesem Fall ist offensichtlich f= S 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadofadaWgaaWcbaGaaGimaaqabaaaaa@3998@ . Wir nehmen daher s>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg6da+iaaicdaaaa@38A3@ an und zeigen nun der Reihe nach:

  1. Es gibt ein wV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiabgIGiolaadAfaaaa@3944@ , mit | w |=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG3baacaGLhWUaayjcSdGaeyypa0JaaGymaaaa@3BC8@ und f(w)=s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bGaaiykaiabg2da9iaadohaaaa@3B27@ .
     
  2. f= S sw MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadofadaWgaaWcbaGaam4CaiaadEhaaeqaaaaa@3AD2@ .

Zu 1.:  Da s das Supremum von M ist, gibt es eine Folge (| f( w n ) |) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaaemaabaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaGaay5bSlaawIa7aiaacMcaaaa@3ECD@ in M, die gegen s konvergiert. Das bedeutet es gibt eine Folge ( w n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3967@ in V mit | w n |=1,   | f( w n ) |s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG3bWaaSbaaSqaaiaad6gaaeqaaaGccaGLhWUaayjcSdGaeyypa0JaaGymaiaacYcacaaMe8+aaqWaaeaacaWGMbGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaaacaGLhWUaayjcSdGaeyizImQaam4Caaaa@4966@ und | f( w n ) |s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaaacaGLhWUaayjcSdGaeyOKH4Qaam4Caaaa@4059@ .

Dabei dürfen wir o.E. annehmen, dass f( w n )0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgwMiZkaaicdaaaa@3CD2@ , also f( w n )s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgsMiJkaadohaaaa@3CFF@ und f( w n )s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgkziUkaadohaaaa@3D37@ , denn ist für ein n der Wert f( w n )<0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgYda8iaaicdaaaa@3C10@ , so kann man w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGUbaabeaaaaa@3804@ durch w n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam4DamaaBaaaleaacaWGUbaabeaaaaa@38F1@ ersetzen.Wir zeigen jetzt: 

( w n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3967@ ist eine Cauchy-Folge. 

Sei dazu ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3952@ vorgegeben, o.E. ε<8 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyipaWJaaGioaaaa@3956@ . Zunächst gibt es ein n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBaaaleaacaaIWaaabeaaaaa@37C2@ , so dass f( w n )s sε 8 =s(1 ε 8 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgwMiZkaadohacqGHsisldaWcaaqaaiaadohacqaH1oqzaeaacaaI4aaaaiabg2da9iaadohacaGGOaGaaGymaiabgkHiTmaalaaabaGaeqyTdugabaGaaGioaaaacaGGPaaaaa@48E6@ für alle n n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3A7B@ . Also hat man  für alle n,m n 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaaaaa@3C1D@ :

f( w n + w m )=f( w n )+f( w m )2s(1 ε 8 ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaam4DamaaBaaaleaacaWGTbaabeaakiaacMcacqGH9aqpcaWGMbGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaGaey4kaSIaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad2gaaeqaaOGaaiykaiabgwMiZkaaikdacaWGZbGaaiikaiaaigdacqGHsisldaWcaaqaaiabew7aLbqaaiaaiIdaaaGaaiykaaaa@5105@ .

Das Parallelogramm-Gesetz erlaubt nun die folgende Abschätzung:
 

| w n w m | 2 =2 | w n | 2 +2 | w m | 2 | w n + w m | 2 4 1 s 2 (f( w n + w m )) 2 4 4 s 2 s 2 (1 ε 8 ) 2 =44+ε ε 2 16 <ε. MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8540@

Als Cauchy-Folge ist ( w n ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadEhadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3967@ konvergent (V ist vollständig!), etwa w n w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGUbaabeaakiabgkziUkaadEhaaaa@3AF7@ . Man hat:

  • | w |=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG3baacaGLhWUaayjcSdGaeyypa0JaaGymaaaa@3BC8@ , denn aus der Abschätzung | w n w |<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iaam4DaaGaay5bSlaawIa7aiabgYda8iabew7aLbaa@3FC4@ gewinnt man über die zweite Dreiecksungleichung
     
    1ε=| w n |ε<| w |<| w n |+ε=1+ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiabew7aLjabg2da9maaemaabaGaam4DamaaBaaaleaacaWGUbaabeaaaOGaay5bSlaawIa7aiabgkHiTiabew7aLjabgYda8maaemaabaGaam4DaaGaay5bSlaawIa7aiabgYda8maaemaabaGaam4DamaaBaaaleaacaWGUbaabeaaaOGaay5bSlaawIa7aiabgUcaRiabew7aLjabg2da9iaaigdacqGHRaWkcqaH1oqzaaa@5459@ ,
    also: | | w |1 |<ε MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaabdaqaaiaadEhaaiaawEa7caGLiWoacqGHsislcaaIXaaacaGLhWUaayjcSdGaeyipaWJaeqyTdugaaa@417C@   für alle ε>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaeyOpa4JaaGimaaaa@3952@ .
     
  • f(w)=s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bGaaiykaiabg2da9iaadohaaaa@3B27@ . Dies folgt mit w n w MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBaaaleaacaWGUbaabeaakiabgkziUkaadEhaaaa@3AF7@ und f( w n )s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgkziUkaadohaaaa@3D37@ aus der Abschätzung:
      
    | f(w)s || f(w)f( w n ) |+| f( w n )s |=| f(w w n ) |+| f( w n )s |s| w w n |+| f( w n )s | MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGMbGaaiikaiaadEhacaGGPaGaeyOeI0Iaam4CaaGaay5bSlaawIa7aiabgsMiJoaaemaabaGaamOzaiaacIcacaWG3bGaaiykaiabgkHiTiaadAgacaGGOaGaam4DamaaBaaaleaacaWGUbaabeaakiaacMcaaiaawEa7caGLiWoacqGHRaWkdaabdaqaaiaadAgacaGGOaGaam4DamaaBaaaleaacaWGUbaabeaakiaacMcacqGHsislcaWGZbaacaGLhWUaayjcSdGaeyypa0ZaaqWaaeaacaWGMbGaaiikaiaadEhacqGHsislcaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaaGaay5bSlaawIa7aiabgUcaRmaaemaabaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgkHiTiaadohaaiaawEa7caGLiWoacqGHKjYOcaWGZbWaaqWaaeaacaWG3bGaeyOeI0Iaam4DamaaBaaaleaacaWGUbaabeaaaOGaay5bSlaawIa7aiabgUcaRmaaemaabaGaamOzaiaacIcacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiabgkHiTiaadohaaiaawEa7caGLiWoaaaa@7EEA@ .

     

Zu 2.:  Wir zeigen jetzt für einen beliebigen, aber festen Vektor xV MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaadAfaaaa@3945@ : f(x)=swx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadohacaWG3bGaey4fIOIaamiEaaaa@3E10@ . Dazu entwickeln wir zunächst für ein α>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOpa4JaaGimaaaa@394A@ die folgende Abschätzung (beachte: | sw |=s=f(w) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGZbGaam4DaaGaay5bSlaawIa7aiabg2da9iaadohacqGH9aqpcaWGMbGaaiikaiaadEhacaGGPaaaaa@4143@  !):

1 α (| swαx || sw |) 1 α ( 1 s | f(swαx) |f(w)) 1 α ( 1 s f(swαx)f(w)) = 1 s f(x) = 1 α ( 1 s f(sw+αx)f(w)) 1 α ( 1 s s| sw+αx || sw |). MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A496@

Also hat man insgesamt:
 

| swαx || sw | α 1 s f(x) | sw+αx || sw | α MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaSaaaeaadaabdaqaaiaadohacaWG3bGaeyOeI0IaeqySdeMaamiEaaGaay5bSlaawIa7aiabgkHiTmaaemaabaGaam4CaiaadEhaaiaawEa7caGLiWoaaeaacqaHXoqyaaGaeyizIm6aaSaaaeaacaaIXaaabaGaam4CaaaacaWGMbGaaiikaiaadIhacaGGPaGaeyizIm6aaSaaaeaadaabdaqaaiaadohacaWG3bGaey4kaSIaeqySdeMaamiEaaGaay5bSlaawIa7aiabgkHiTmaaemaabaGaam4CaiaadEhaaiaawEa7caGLiWoaaeaacqaHXoqyaaaaaa@5FDB@ .

Das Ableitungsverhalten der Funktionen φ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdygaaa@37A2@ und ψ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdKhaaa@37B7@ aus der Vorbemerkung liefert nun die gewünschte Gleichheit:
 

swx=s(sw)°x=sφ'(0)f(x)sψ'(0)=s(sw)°x=swx MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaadEhacqGHxiIkcaWG4bGaeyypa0Jaam4CaiaacIcacaWGZbGaam4DaiaacMcacqGHWcaScqGHxiIkcaWG4bGaeyypa0JaeyOeI0Iaam4CaiabeA8aMjaacEcacaGGOaGaaGimaiaacMcacqGHKjYOcaWGMbGaaiikaiaadIhacaGGPaGaeyizImQaam4CaiabeI8a5jaacEcacaGGOaGaaGimaiaacMcacqGH9aqpcaWGZbGaaiikaiaadohacaWG3bGaaiykaiabgclaWkabgEHiQiaadIhacqGH9aqpcaWGZbGaam4DaiabgEHiQiaadIhaaaa@6486@ .

 

 


 9.13
9.15.