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By G. Coeuré (Eds.)

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

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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 ?

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