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By Pierre Mazet

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Example text

The case of integral rings. The following propositions enable us to prove, by induction, that an integral ring is n-Noetherian o r C . M . n . , . Proof Let A be an integral ring. Every nonzero ideal i n contains a regular sequence o f length 0-dense; it follows from this that 0 I and A Furthermore, since ht P \< ht P' we have corollary to proposition 3 . 8. , A-module Here, since is of finite type over A-module an is a n-Noetherian ring. We d u p p o b e t h a t grP' . M. 0-closed ideals and therefore that 39 A is 0-Noetherian.

C this definition coincides with the usual definition of a n analytic function. The following two propositions show that we c a n always restrict attention to this case. 3. Let 51 be a n o p e n s u b ’ s e t of a l o c a l l y c o n v e x s p a c e U 9 o r d e r t h a t a map U from E to be . E G-anaZytic, In i t is n e c e s s a r y and s u f f i c i e n t t h a t i t s r e s t r i c t i o n t o e a c h a f f i n e - t h a t i s t o s a y , for a l l S i n e be holomorphic t t h e map definition.

This Page Intentionally Left Blank 49 CHAPTER 1 ANALYTIC MAPS The extension of the notion o f analytic map to spaces o f infinite dimension poses certain problems o f definition. are studied in many works, for example [ BS] and These problems [NB] . The natural idea o f considering maps that c a n be represented as a series o f polynomials leads to the notion o f Gtteaux analytic maps; however these need not be continuous, and we are thus led to add the hypothesis o f continuity in order to obtain a "good" theory.

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