# Download Analytic Functions and Manifolds in Infinite Dimensional by G. Coeuré (Eds.) PDF

By G. Coeuré (Eds.)

Coeure G. Analytic services and manifolds in endless dimensional areas (NHMS, NH, 1974)(ISBN 0444106219)

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Extra info for Analytic Functions and Manifolds in Infinite Dimensional Spaces

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F l l w , If t h e boundary of w has a s i n g u l a r p o i n t f o r simultaneous continua- r , then t i o n of l' i s not s t r o n g l y i n v a r i a n t . Actually, suppose t h e conver- se ; we should have : Given for all 1. k k r( as r a d i u s of convergence. This i s impossible n e a r a s i n g u l a r p o i n t o f t h e boundary o f w. We can t a k e f o r w singular point f o r exp , the h a l f plane Re z < 0 , where the origin is a which belongs t o 'I , W e study t h e e x i s t e n c e o f some type o f Frechet spaces The a d j o i n t space E' logy induced by t h a t of We say of i n @x(X,C).

The t o p o l o g y by . Let T€ be g i ve n and K(X) AT b e t h e spa c e spanned and normed by t h e Minkowski-norm a s s o c i a t e d w i t h T mapping AT -). - We have . AT h T . The c a n o n i c a l i s contained i n is d i r e c t e d by set i n c l u s i o n . b) and the e q u a l i t y of p o i n t w i s e convergence and precompact convergence on any e q u i c o n t i n u o u s s e t . € o r t h e l e f t hand, w e have only t o check t h e c o n t i n u i t y of e a c h mapping AT + 5 ( X g .

For any open, b a l a n c e d , neighbourhood in X , the and E is a metrizable OE of V E , and any s e t f o l l o w i n g 'tboundaryt' f u n c t i o n s are d e f i n e d : V d (x) X V d (TI X SUP {r >/ b inf dX(x) [ %*+-rV V x c T is c o n t a i n e d i n X 1 . Definition 4 . 3 . - Let r be a s e t i n ~,(x,z); t h e s e t T f o r r i s defined by : 3 r i = { X E x [ [ [ f ( z ~ lll lf l ~ l , aZZ T-huZZ o f a bounding f cr 1 . Let 'l be a e. i. d, natura2 Frechet space i n 6xt~,~). Then T i n the I'-maximaZ extension ?