L(E) est l′algebre des endomorphisme continus d′un espace de Banach E, muni de la norme d′application lineaire ; GL(E) est le sous ensemble des elements inversibles de L(E) a)Montrer que GL(E) est ouvert dans L(E) b)montrer que l′application ∅:GL(E)→GL(E) u→u^(−1) =∅(u) est continu dans GL(E) c)montrer que ∅ est differentiable dans GL(E) dt calculer d∅


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Question Number 176482 by Vynho last updated on 19/Sep/22

$L (E) e s t l^{'} a l g e b r e d e s e n d o m o r p h i s m e$ $c o n t i n u s d^{'} u n e s p a c e d e B a n a c h E,$ $m u n i d e l a n o r m e d^{'} a p p l i c a t i o n l i n e a i r e$ $; G L (E) e s t l e s o u s e n s e m b l e d e s$ $e l e m e n t s i n v e r s i b l e s d e L (E)$ $a) M o n t r e r q u e G L (E) e s t o u v e r t d a n s$ $L (E)$ $b) m o n t r e r q u e l^{'} a p p l i c a t i o n$ $\emptyset : G L (E) \to G L (E)$ $u \to u^{- 1} = \emptyset (u) e s t c o n t i n u d a n s G L (E)$ $c) m o n t r e r q u e \emptyset e s t d i f f e r e n t i a b l e$ $d a n s G L (E) d t c a l c u l e r d \emptyset$
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