Kompakt erzeugte Vektorräume und Analysis by Ulrich Seip (auth.) PDF

Kompakt erzeugte Vektorräume und Analysis by Ulrich Seip (auth.) PDF

By Ulrich Seip (auth.)

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3) Ist Q~Mor~(Vl,V2) und A:V2--~V 3 linear, ~ A ~ Q ) =kO~Q :VlX Vi× R --+V 2 --~ V 3 . so gilt f~r ~(kaQ): 50 4) Ist QI~MOr~(VI,V2 ) , A:VI--~ V 2 linear und Q2eMor~(V2,V3) , so gilt: VI×V1 x K -~ VI~VI×VlXV1 × RxR --~ Vl× Vl× V2×V2× R -~ V2×V2xV2 × R -~ V 2 × V 2 × R --~ V 3 . Wir definieren jetzt den Vektorraum zeugte Vektorr~ume, Produkte wobei wir beachten, Mit S(EnEnR,F) der Form (0,y,0) als Teilmenge bezeichnen stetig rators ~ zeigt, Definition entscheidenden am Nullpunkt ab. Die Komposition lineare Abbildung Nullabbildung nach ist.

51 Wir kommen Hierbei zum D i f f e r e n z i e r b a r k e i t s b e g r i f f von F u n k t i o n e n ist O eine m i t der T e i l r a u m t o p o l o g i e v o n E u n d wir w e r d e n (nicht n o t w e n d i g pakt erzeugter Vektorr~ume vektorieller Unterr~ume) kom- vorgelegt. Dann heisst ~ an verwenden. Es sei e i n e F u n k t i o n der S t e l l e x ~ E d i f f e r e n z i e r b a r , o A : E - - ~ F gibt, offene Teilmenge d e n B u c h s t a b e n O v o n n u n an n u r n o c h zur B e z e i c h - nung o f f e n e r T e i l r ~ u m e 5 DEFINITION versehene ~:E=O--~F.

Das neue a:EmOl---~F ist daher ein HomSomor- einmal Ra -I(0)=G -I-(D~(O))-I:F=O2--~E. Wir setzen Damit erhalten wit auf O1: =(Da(O ))-lo~+R -1(0)~ ~=(Da(O))-lo{Da(0)+Ra(D) =I+(D~(0))-loR~(0)+Ra-I(0)~ }+Ra -1(0)~6= 6. Daraus folgt: R~ -1(0) =-(Da(0))-I~R~(0)~ 6-1 :F=O2 --, E. Nach Voraussetzung fast-lipschitz am Nullpunkt. ist -1 Es existiert daher eine Funktion • :FnFnR~(~F~(lmPF))-Io2~(O2nFnR)--~E, (0,Y2,0) 0/ inverse Funktionen phismus yon O 1 auf 02, und da 6(0)=0, gilt auch ~-I(0)=0. 101= -1 und D~(Xo) die stetig an den Punkten ist und fur die ~(yl,Y2,r)r=a-l(yl+Y2r)-a-l(yl) gilt.

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