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# Complex Analysis in Banach Spaces: Holomorphic Functions and by Jorge Mujica

By Jorge Mujica

Difficulties coming up from the learn of holomorphic continuation and holomorphic approximation were crucial within the improvement of complicated research in finitely many variables, and represent some of the most promising strains of analysis in endless dimensional complicated research. This ebook provides a unified view of those themes in either finite and endless dimensions.

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Example text

B) The r a d i u s of convergence o f t h e T a y l o r series of f a t a w i l l b e d e n o t e d by r e f l a ) . For s h o r t r c f ( a ) w i l l be r e f e r r e d t o as t h e r a d i u s of c o n v e r g e n c e o f f a t a . (c) The d i s t a n c e from ci t o t h e boundaryof U w i l l b e den o t e d by d u ( a ) . When U = E t h e n f o r convenience w e d e f i n e d , ( a ) = m. 7 . 1 3 THEOREM.

E n ) be a b a s i s f o r E and l e t 5, 5, d e n o t e t h e c o r r e s p o n d i n g c o o r d i n a t e f u n c t i o n a l s . K. Let is taken over multi-indices all v. such t h a t la1 = m. E iTz P ( m E ; F ) d e n o t e t h e s u b s p a c e of f all P E P(mE;F) which c a n b e w r i t t e n i n t h e form where c i E F and sum of t h e s p a c e s E E'. Let P ( m E ; F ) with f P ( E ; F ) denote t h e a l g e b r a i c f n o . Each m E i s s a i d t o be a c o n t i n u o u s poZynorniaZ F = LK f P (E;M) = P (E).

C) P i s bounded on some open b a l l . 0. z i s c o n t i n u o u s i f and o n l y JI o P E P a ( E I i s c o n t i n u o u s f o r e v e r y i f t h e polynomial d, E F'. 3. L that i f polynomial of d e g r e e a t most rn t h e n P E PafE;F) is a P i a + Xb) i s a p o l y n o m i a l i n X o f d e g r e e a t most m f o r a l l a , b E E . I n t h i s s e c t i o n w e show t h a t t h e c o n v e r s e i s a l s o t r u e , t h u s p r e s e n t i n g an a l t e r n a t i v e d e s c r i p t i o n of p o l y n o m i a l s i n Banach s p a c e s .