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F i r s t we quote t h e s o u r c e of t h e t h e o r e m s p r e s e n t e d i n t h e book, a n d t h e n we give s o m e m o r e r e f e r e n c e s of p r e v i o u s o r r e l a t e d r e s u l t s . T h e r e f o r e n u m e r o u s i m p o r t a n t p a p e r s c o n n e c t e d with t h e t o p i c s d i s c u s s e d a r e not m e n t i o n e d . CHAPTER 1 T h e o r e m s 1 . 4 .7-8 w e r e obtained i n C o l o m b e a u [3,17] a n d t h e o r e m s 1 . 6 . 3 w a s obtained in C o l - m b e a u - M u j i c a [l] . T h e concept of differentiable m a p p i n g s in t h e s e n s e of 8 1.1 a n d 6 1 . 2 i s i n Sebastigo e Silva [l, 21 in t h e c a s e of t h e Von N e u m a n n b o r nologies of l o c a l l y convex s p a c e s . T h e n h e p l a c e d h i s t h e o r y i n i t s n a t u r a l s e t t i n g of bornological v e c t o r s p a c e s in S e b a s t i c o e Silva [3]. T h e c o n Cco m a p p i n g s in the e n l a r g e d s e n s e c o n s i d e r e d in $ 1 . 4 w a s i n c e p t of t r o d u c e d in Sebastia40 e Silva [ Z ] . T h e concept of d i f f e r e n t i a b l e m a p p i n g s in t h e s e n s e of $ 1.1 a n d 8 1 . 2 w a s a l s o i n t r o d u c e d i n M a r i n e s c u [l] in the s e t t i n g of bornological v e c t o r s p a c e s ( p o l y n o r m e d s p a c e s i n M a r i n e s c u [ l ] ' s t e r m i n o l o g y ) . T h e r e i s a slight d i f f e r e n c e b e t w e e n M a r i n e s c u ' s definition a n d t h e definition in $ 1.1, 0 1 . 2 , s e e C o l o m b e a u [ 2 ] , p . 20 a n d p.31-32. F o r f u r t h e r c o m p a r i s o n r e s u l t s between t h e t h r e e c o n c e p t s of differentiability of 3 1 . 1 , 0 1 . 2 , 8 1 . 4 a n d $ 1. 5 we r e f e r to C o l o m b e a u [ 3 , 4,171 . F o r the g e n e r a l l i t t e r a t u r e on d i f f e r e n t i a b l e m a p p i n g s between l o c a l l y convex s p a c e s we r e f e r t o t h e e x c e l l e n t s u r v e y s of A v e r b u c k Smolyanov [ Z ] a n d N a s h e d [l] t h a t contain e x c e l l e n t b i b l i o g r a p h i e s f o r p a p e r s a n t e r i o r to 1969. Among t h e v e r y n u m e r o u s books a n d a r t i c l e s on t h i s s u j e c t l e t u s f u r t h e r m o r e quote Averbuck-Smolyanov [l] , B u c h e r F r b l i c h e r [l] , K e l l e r [l] , Y a m a m u r o [l, 21, M e i s e [l], C o l o m b e a u Meise [l]. In t h i s book we did n o t c o n s i d e r t h e v e r y i m p o r t a n t t o p i c s of I m p l i c i t F u n c t i o n s a n d O r d i n a r y D i f f e r e n t i a l E q u a t i o n s . In t h e c a s e of Banach s p a c e s v e r y g e n e r a l r e s u l t s w e r e obtained a l r e a d y i n t h e t h i r t i e s ( s e e C a r t a n [l] o r DieudonnC [l] f o r i n s t a n c e ) a n d a r e now v e r y c l a s s i c a l . In the c a s e of non n o r m e d s p a c e s t h e s i t u a t i o n i s c o n s i d e r a b l y m o r e c o m p l i c a t e d a n d does not s e e m t o be c o m p l e t e l y c l a r i f i e d at p r e s e n t . So we j u s t s k e t c h h e r e t h i s topic a n d give s o m e r e f e r e n c e s . L e t u s f i r s t a s s u m e the e x i s t e n c e of a continuous i m p l i c i t f u n c tion o r of a continuous i n v e r s e , when t h e g i v e n f u n c t i o n s a r e d i f f e r e n t i a b l e In t h e Banach s p a c e c a s e t h i s i m p l i c i t function o r t h i s i n v e r s e m a p i s diff e r e n t i a b l e (Nachbin [43, P r o p o s i t i o n 16.16 a n d 22.10 f o r i n s t a n c e ) . A c o u n t e r e x a m p l e i n Averbuck-Smolyanov [ 2 ) shows that t h i s is no l o n g e r t r u e in g e n e r a l l o c a l l y convex s p a c e s . On t h i s topic s e e C o l o m b e a u [ 4 ] , Smolyanov [l]. If one a s s u m e s s o m e p r o p e r t y s t r o n g e r t h a n continuity on t h e i m p l i c i t o r i n v e r s e m a p , one o b t a i n s the d e s i r e d differentiability 423
Bibliographic Notes
424
r e s u l t , s e e C o l o m b e a u [4], t h e o r e m 2 . 2 which i s q u i t e g e n e r a l . Now l e t i s a n - t i m e s d i f f e r e n t i a b l e b i j e c t i o n a n d t h a t its i n v e r s e m a p us assume f f - l i s one t i m e d i f f e r e n t i a b l e ( s a m e s i t u a t i o n a n d r e s u l t s i n t h e c a s e of t h e i m p l i c i t f u n c t i o n ) . In t h e c a s e of B a n a c h s p a c e s f - l i s n - t i m e s d i f f e rentiable. A counterexample in Colombeau [4], $ 6 , example 2 , shows t h a t t h i s i s no l o n g e r t r u e i n g e n e r a l i n non n o r m e d s p a c e s , but t h e o r e m 3 . 2 i n C o l o m b e a u [4] g i v e s a v e r y g e n e r a l c a s e i n which f - l i s n - t i m e s d i f f e r e n t i a b l e . In s h o r t t h e r e s u l t s a r e q u i t e g e n e r a l , but one n e e d s t o a s s u m e m o r e r e s t r i c t i v e a s s u m p t i o n s t h a n in t h e B a n a c h s p a c e s c a s e (the continuity of t h e i m p l i c i t f u n c t i o n i m p l i e s t h e s e p r o p e r t i e s i n t h e B a n a c h s p a c e s c a s e ) . The s i t u a t i o n of t h e e x i s t e n c e r e s u l t s i s m u c h w o r s e : L e t u s now c o n s i d e r t h e p r o b l e m of e x i s t e n c e of a n i m p l i c i t f u n c t i o n o r of a n i n v e r s e of a g i v e n d i f f e r e n t i a b l e m a p , in t h e c o n d i t i o n s t h a t , i n B a n a c h s p a c e s , e n s u r e t h e i r e x i s t e n c e ( C a r t a n [ l ] , Dieudonng [l], . ) . C o u n t e r e x a m p l e s i n E e l l s [l] , P i s a n e l l i [l] , C o l o m b e a u [l] , L o j a s i e w i c z J r . [l] show t h a t t h e v e r y n e a t r e s u l t s of t h e B a n a c h s p a c e s f r a m e w o r k do not r e m a i n v a l i d i n l o c a l l y c o n v e x s p a c e s . T h e r e is a v e r y l a r g e a m o u n t of w o r k s on t h i s t o p i c : people f i r s t t r i e d v a r i o u s g e n e r a l i z a t i o n s of the c l a s s i c a l p r o o f , s e e f o r i n s t a n c e F a l b - J a c o b [l], M a c D e r m o t t [l, 2 , 31, C o l o m b e a u [l, 5, 6 , 7 1 , Y a m a m u r o [ 2 ] , T h e i m p l i c i t function a n d l o c a l i n v e r s i o n t h e o r e m s i n C o l o m b e a u [l, 5, 61 a r e u s e d i n J . A . L e s l i e [ 3 ] f o r a proof of a K u p k a - S m a l e t h e o r e m in t h e r e a l a n a l y t i c c a s e . A d i f f e r e n t m e t h o d w a s i n s p i r e d by t h e " N a s h I m p l i c i t F u n c t i o n T h e o r e m " , s e e J . T . S c h w a r z [l], M o s e r [l] , H a m i l t o n [l] , S e r g e r a e r t [l] , J a c o b o w i t z [ l ] , Z e h n d e r [l], L o j a s i e w i c z - Z e h n d e r [l] T h i s o t h e r kind of i m p l i c i t function t h e o r e m s have c l a s s i c a l p o w e r f u l a p p l i c a t i o n s , f o r i n s t a n c e in t h e proof of t h e e m b e d d i n g of R i e m a n n i a n m a n i f o l d s i n IR" , s e e S c h w a r z [l]
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Now l e t us c o n s i d e r " o r d i n a r y " d i f f e r e n t i a l e q u a t i o n s X ' ( t ) = F ( X ( t ) ,t ) , X(to) = Xo ( C a u c h y p r o b l e m s ) . T h e v e r y b e a u t i f u l e x i s t e n c e , u n i q u e n e s s a n d d e p e n d e n c e on p a r a ' m e t e r s a n d d a t a r e s u l t s of t h e B a n a c h s p a c e c a s e t h e o r y ( C a r t a n [l] , Dieudonne [l] , . ) do not r e m a i n v a l i d i n l o c a l l y convex s p a c e s , s e e C o l o m b e a u [2] , P l i s [l], De G i o r g i [l]). V a r i o u s m e t h o d s w e r e u s e d : a c o m p a c t n e s s m e t h o d in Dubinsky [l], a n , i t e r a t i o n m e t h o d in C o l o m b e a u 61, 2 , 61, T r e v e s [2] , L e s l i e [2], V e r y good r e s u l t s a n d a p p l i c a t i o n s c a m e f r o m a v e r y s t r o n g r e i n f o r c e m e n t of t h i s i t e r a t i o n m e t h o d by a n a s t u t e m a j o r i z a t i o n t e c h n i q u e i n t h e s o c a l l e d "Ovcyannikov method" : Ovcyannikov [l, 2 3 , T r e v e s [ 3 , 4 , 51, S t e i m b e r g T r e v e s [l], S t e i m b e r g [l], P i s a n e l l i [2], N i r e m b e r g [l], Du C h a t e a u [l], L a s c a r [2], . T h e s e r e f e r e n c e s c o n c e r n t h e s t u d y at o r d i n a r y points (for a c l a s s i f i c a t i o n of s i n g u l a r i t i e s of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s s e e Wasow [ l ] ) . T h e Ovcyannikov m e t h o d w a s u s e d i n t h e c a s e of " r e g u l a r s i n g u l a r points" i n Baouendi-Goulaouic [l, 21 a n d i n t h e c a s e of " i r r e g u l a r s i n g u l a r points" i n C o l o m b e a u - M C r i l [l]
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B e s i d e s the a b o v e t o p i c s , D i f f e r e n t i a l C a l c u l u s i n a . c . s . h a s a l o t of v a r i o u s a p p l i c a t i o n s , m o s t of t h e m being at p r e s e n t i n f u l l d e v e l o p -
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m e n t : f o r i n s t a n c e d i f f e r e n t i a l s t r u c t u r e s i n s e t s of COD o r real a n a l y tic d i f f e o m o r p h i s m s of c o m p a c t R i e m a n n i a n m a n i f o l d s ( L e s l i e [l, 31 ) o r new c o n c e p t s of g e n e r a l i z e d f u n c t i o n s on IE?? , giving a m e a n i n g t o a n y p r o d u c t of d i s t r i b u t i o n s ( C o l o m b e a u [l8] ), j u s t t o quote a few of t h e m . CHAPTER 2 T h e o r e m 2.2.3 w a s obtained in L a z e t [l, 2 3 . T h e o r e m s 2 . 3 . 3 - 4 a r e i n C o l o m b e a u [ 8 ] . T h e o r e m 2 . 4 . 1 and c o r o l l a r i e s w e r e o b t a i n e d i n L a z e t [l, 21, C o l o m b e a u [9,10] T h e c o u n t e r e x a m p l e in 2 . 5 w a s obtained i n C o l o m b e a u [9,10] . T h e o r e m 2 . 6 . 4 w a s o b t a i n e d i n B o c h n a k - S i c i a k [ l , 23 a n d T h e o r e m 2 . 7 . 4 i n C o l o m b e a u - M u j i c a [l]
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T h e concept of Silva h o l o m o r p h i c m a p p i n g s w a s i n t r o d u c e d in S e b a s t i s e Silva [l, 2 , 33, a n d t h e n s t u d i e d i n m o r e d e t a i l s in C o l o m b e a u [ 9 , 1 0 , 8 , 1 , 1 7 ] , C o l o m b e a u - L a z e t [l], L a z e t [l, 23, P i s a n e l l i [3, 43, M a t o s Nachbin [l] , B i a n c h i n i [l] , B i a n c h i n i - P a q u e s - Z a i n e [l], e t c . L e t u s m e n tion a n i c e o r i g i n a l i n t r o d u c t i o n i n M e i s e - V o g t [l] a n d t h a t s e v e r a l a u t h o r s i n t r o d u c e d t h e concept of "hypo-analytic m a p p i n g s " which c o i n c i d e s with t h e one of Silva h o l o m o r p h i c m a p p i n g s i n all "usual" A . c s . .
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T h e c o n c e p t of h o l o m o r p h i c ( = G - a n a l y t i c a n d c o n t i n u o u s ) m a p p i n g s i n l o c a l l y convex s p a c e s h a s b e e n s t u d i e d by a c o n s i d e r a b l e n u m b e r of a u t h o r s , a n d we r e f e r t o D i n e e n ' s r e c e n t book [I] f o r r e f e r e n c e s , a s w e l l a s t o Nachbin [l] f o r a n i n t r o d u c t i o n a n d r e f e r e n c e s . T h e r e a r e a l s o m a n y books a n d v o l u m e s of P r o c e e d i n g s c o n c e r n i n g t h i s s u b j e c t : B a r r o s o [l, 2 , 3, 41, Boland [3], C o e u r 6 [l], Dineen [l] , Hayden-Suffridge [l] , HervC [l], Lelong [ Z , 33, Lelong-Skoda [l], Machado [l] , M a t o s [7], Mazet [l] , Mujica [l] , Nachbin [ Z ] , N o v e r r a z [l, 43, Z a p a t a [l] , e t c . L e t u s a l s o m e n t i o n the s u r v e y a r t i c l e s N a c h b i n [l, 3 , 51, B i e r s t e d t - M e i s e [Z] , C o l o m b e a u - M a t o s [ Z ] , Dineen [lo] Concerning holomorphic functions on n u c l e a r F r k c h e t s p a c e s a n d v e r y i m p o r t a n t c o u n t e r e x a m p l e s s e e M e i s e Vogt [ 2 , 31
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T h e h o l o m o r p h i c r e p r e s e n t a t i o n of F o c k s p a c e s of B o s o n f i e l d s i s e x p o s e d o r u s e d i n B e r e z i n [l] , Novozhilov-Tulub [l], S a r a v a s t i - V a l a t i n [l], R z e w u s k i [l, 2 3 , Dwyer [l, 2 , 3 , 4 J , C o l o m b e a u - P e r r o t [l, 2 , 31, K r 6 e [l, 21, K r g e - R a c z k a [l]
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CHAPTER 3 T h e o r e m s 3 . 1 . 2 , 3 . 2 . 1 , 3 . 2 . 3 a n d 3 . 3 . 1 a r e r e f o r m u l a t i o n s of c l a s s i c a l r e s u l t s . E x a m p l e 3 ' 2 . 4 is t a k e n f r o m C o l o m b e a u [9,10] ; a s i milar c o u n t e r e x a m p l e w a s o b t a i n e d independently in H i r s c h o w i t z [l] E x a m p l e 3 . 3 . 3 i s t a k e n f r o m Colombeau [17] A m o r e g e n e r a l f o r m of t h e o r e m 3 . 4 . 3 - 4 i s in C o l o m b e a u - L a z e t [l] , C o l o m b e a u [lo]
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S o m e l i t t e r a t u r e on Z o r n , H a r t o g s a n d M o n t e l ' s t h e o r e m s i s i n Dineen [ l ] , S e b a s t i c o e Silva [l, 21, Hille [l] , H i l l e - P h i l l i p s [l],
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L a z e t [l, 2 3 , Colombeau [l, 9,10,17], Col.ombeau-Lazet [l] , N o v e r r a z [l, 2 1 , B o c h n a k - S i c i a k [l, 21, P i s a n e l l i [3, 43, Matos [I, 23, .
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CHAPTER 4 T h e o r e m s 4 . 2 . 1 - 2 , 4 . 3 . 1 , 4 . 4 . 1 w e r e obtained i n C o l o m b e a u P e r r o t [2,4,5]. T h e o r e m s 4 . 2 . 4 , 4 . 3 . 4 , 4 . 4 . 2 w e r e obtained i n a n unpublished m a n u s c r i p t Colombeau [ll] ; s e e a l s o M e i s e [l] a n d C o l o m b e a u M e i s e [l]. T h e s t r i c t l y c o m p a c t p o r t e d topology w a s i n t r o d u c e d i n BianchiniP a q u e s - Z a i n e [l] a n d p r o p o s i t i o n 4 . 1 . 4 w a s obtained i n C o l o m b e a u - M e i s e P e r r o t [l] N a c h b i n ' s p o r t e d t o p o l o g y w a s defined i n Nachbin [2] f o r hol o m o r p h i c functions on Banach s p a c e s . T h i s last book o r i g i n a t e d a v e r y i m p o r t a n t t r e n d of w o r k s , in p a r t i c u l a r on topologies i n s p a c e s of holom o r p h i c m a p p i n g s , a n d t h i s last topic i s t r e a t e d i n d e t a i l i n t h e r e c e n t book Dineen [l], to which we r e f e r . Let u s j u s t m e n t i o n r e c e n t r e s u l t s i n Dineen [9], Boland-Dineen [4], M e i s e [ 21, Mujica €21
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CHAPTER 5 T h e o r e m 5.1.2 w a s obtained i n the m o r e g e n e r a l c a s e of a f i n i t e l y Runge open s e t 0 in C o l o m b e a u - M e i s e - P e r r o t [l]. T h e o r e m 5 . 2 . 1 w a s obtained in C o l o m b e a u - M e i s e [l], s e e a l s o M e i s e [l] T h e proof of t h e o r e m 5 . 3 . 1 is a n i m m e d i a t e a d a p t a t i o n of a proof i n B o n i c - F r a m p t o n [I].
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G e n e r a l i z a t i o n s of N a c h b i n ' s A p p r o x i m a t i o n T h e o r e m a r e i n P r o l l a [ 2 , 31, G u e r e i r o - P r o l l a [l], s e e a l s o Nachbin [7]
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T h e r e s u l t s in t h i s c h a p t e r w e r e c h o s e n s i n c e we u s e t h e m i n t h i s book. S e v e r a l o t h e r r e s u l t s of e x i s t e n c e a n d a p p r o x i m a t i o n a r e i n c h a p t e r s 6 , 9 t o 16. S o m e r e s u l t s which a r e not i n t h i s book a r e s u r v e y e d i n : Dineen [l], J o s e f s o n Schottenloher [l] T h e y c o n c e r n (1) . , ( 2 ) e x i s t e n c e of ho[ 1 , 2 3 , R u s e k [I], Bayoumi [l], l o m o r p h i c functions with p r e s c r i b e d r a d i u s of c o n v e r e n c e : A r o n T i f , C o e u r 6 [2], K i s e l m a n 11, 2 , 31, Schotten-&rvey in Schottenloher [l] contains a l s o o t h e r t o p i c s r e l a t e d with following c h a p t e r s of t h i s book. E x i s t e n c e r e s u l t s on h o l o m o r p h i c f u n c t i o n s with p r e s c r i b e d v a l u e s a t a n infinite given s e t of points a r e in H e r v i e r [I], Valdivia [l] R e s u l t s on e x t e n s i o n s of h o l o m o r p h i c f u n c t i o n s defined in a c l o s e d s u b s p a c e of a l o c a l l y convex s p a c e a r e i n B o l a n d r 3 ] , A r o n - B e r n e r [l], Colombeau-Mujica Meise-Vogt [3] R e s u l t s of e x i s t e n c e of Co3 holomorphic ma in s with p r e s c r i b e d a s y m p t o t i c e x a n s i o n s a r e i n G b e a T d k & - 5 , 1 6 ] , Colombeau-Mujici&zr.
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CHAPTER 6 T h e o r e m 6.1.1 w a s obtained in a m o r e r e s t r i c t i v e c a s e i n A r o n Schottenloher [l], Schottenloher [3], a n d in m o r e g e n e r a l i t y i n P a q u e s [l]
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Bibliographic Notes
C o r o l l a r y 6.1.4 w a s d i r e c t l y obtained f o r Cn f u n c t i o n s in M e i s e [l] w h e r e a d e t a i l e d proof is g i v e n . A m o r e g e n e r a l f o r m of t h e o r e m 6 . 2 . 1 i s in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 5 . C o r o l l a r y 6 . 2 . 2 w a s d i r e c t l y obtained in Colombeau [ll] a n d i s a l s o in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 6 T h e o r e m 6 . 3 . 2 i s i n C o l o m b e a u - M e i s e - P e r r o t [l] A m o r e g e n e r a l f o r m of t h e o r e m 6 . 3 . 3 i s in C o l o m b e a u - M e i s e [l] r e m a r k 4 . 5 .
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F o r o t h e r p a p e r s on t h e A p p r o x i m a t i o n P r o p e r t y of 'Q (Q), X(Q) a n d Cn(Q) s e e P a q u e s [2], A r o n - S c h o t t e n l o h e r [l] , M e i s e [l], B o m b a l Gordon-Gonzalez Llavona [l] . U s e of the " k e r n e l t h e o r e m " 6.1.4 t o a study of l i n e a r o p e r a t o r s o n F o c k s p a c e s of Boson f i e l d s i s done in C o l o m b e a u - P e r r o t [l, 2 , 3 ] , Kre'e [l, 21, K r i e - R a c z k a [l]
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CHAPTER 7 T h e o r e m 7 . 2 . 1 w a s obtained in C o l o m b e a u - P e r r o t [6] but is a l s o a c o n s e q u e n c e of a t h e o r e m in Boland (2, 31, s e e C o l o m b e a u - P e r r o t [6]. T h e o r e m 7 . 4 . 1 is i n A n s e m i l - C o l o m b e a u [l] a n d t h e o r e m " 7 . 4 . 7 i n C o l o m b e a u - P o n t e [l]
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T h e f i r s t r e s u l t o n t h e F o u r i e r - B o r e 1 i s o m o r p h i s m i n infinite d i m e n s i o n w a s obtained by Gupta [l, 2 , 3 ] f o r e n t i r e f u n c t i o n s of n u c l e a r type on Banach s p a c e s . T h i s r e s u l t w a s e x t e n d e d to l o c a l l y convex s p a c e s in M a t o s [ 3 , 4 ] a n d Boland [ 2 , 3 ] . F o r connections b e t w e e n t h e above r e s u l t s of Boland a n d Matos s e e C o l o m b e a u - M a t o s [l] . S p a c e s of h o l o m o r p h i c g e r m s w e r e i n t e n s i v e l y s t u d i e d , s e e t h e book Dineen [l] ; let u s only quote Mujica [l. 2, 31, Dineen [5], B i e r s t e d t M e i s e (1, 21, A r a g o n a [l] , S o r a g g i [l] , Biagioni [l]
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T h e difficulty to extend the c l a s s i c a l P a l e y - W i e n e r - S c h w a r t z t h e o r e m t o infinite d i m e n s i o n w a s pointed out i n Dineen-Nachbin [l] T h e n a P. W .S. t h e o r e m w a s obtained i n A b u a b a r a [l, 21 a n d h i s i d e a w a s a d a p t e d t o the c a s e of n u c l e a r s p a c e s i n A n s e m i l - C o l o m b e a u [l]. O t h e r r e s u l t s w e r e obtained by defining a s u i t a b l e s p a c e of Coo f u n c t i o n s i n C o l o m b e a u Ponte [l] f o r t h e c a s e of n u c l e a r s p a c e s a n d in C o l o m b e a u - P a q u e s [l] f o r the c a s e of Banach s p a c e s . See a l s o o t h e r kinds of P . W . S . t h e o r e m s in t h i s book c h a p t e r 13 ( C o l o m b e a u - P a q u e s [3] ), i n C o l o m b e a u - P a q u e s [a] a n d Gal6 [l]
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CHAPTER 8 T h e o r e m s 8.1.1-2 a r e r e f o r m u l a t i o n s of a r e s u l t obtained i n d e pendently i n Boland [4] a n d W a e l b r o e c k [I] T h e proof g i v e n h e r e is t h a t of C o l o m b e a u - P e r r o t [ 7 ] obtained l a t e r . T h e o r e m s 8 . 2 . 5 - 6 w e r e g i v e n in C o l o m b e a u - M e i s e [2] a n d a r e e x t e n s i o n s of a r e s u l t i n B i e r s t e d t - G r a m s c h Meise [l] T h e o r e m 8 . 3 . 2 i s due to M e i s e [ 3 ] , s e e a l s o C o l o m b e a u - M e i s e
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bl
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428
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M o r e r e c e n t l y in B o r g e n s - M e i s e - V o g t [l, 2, 31 a p p e a r e d a unified proof of all n u c l e a r i t y r e s u l t s p r e s e n t l y known in Infinite D i m e n s i o n a l Holomorphy, a n d i t is e a s y to s e e t h a t t h e m e t h o d s t h e r e a l s o apply t o the s p a c e s K s ( n ) s i n c e the p r o o f s r e l y on l e m m a s dealing with t h e B a n a c h space situation. CHAPTER 9 T h e o r e m 9 . 4 . 1 i s in C o l o m b e a u - P e r r o t [8], a s well a s a p h y s i c a l i n t e r p r e t a t i o n of t h e s e e q u a t i o n s . A d i f f e r e n t proof a n d f u r t h e r r e s u l t s a r e in Colombeau-Matos [2]. CHAPTER 10 T h e o r e m s 1 0 . 2 . 6 and 1 0 . 3 . 4 a r e i n C o l o m b e a u - P e r r o t [9] a n d we expose h e r e the proof of t h i s p a p e r s i n c e we s h a l l n e e d it a l s o i n c h a p t e r 16. Another proof will be given in c h a p t e r 11, a n d will be t a k e n f r o m C o l o m b e a u G a y - P e r r o t [l] A t h i r d proof is i n C o l o m b e a u - M a t o s [ 2 ] .
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O t h e r r e s u l t s on convolution e q u a t i o n s i n s p a c e s of e n t i r e f u n c t i o n s of exponential type a r e i n Boland-Dineen [3] ; they a r e obtained f r o m t h e f i n i t e d i m e n s i o n a l r e s u l t s by a method of t r a n s f i n i t e induction. A g r e a t a m o u n t of r e s u l t s on r e l a t e d e q u a t i o n s i s i n Dwyer [l, 2, 3, 4,5, 6, 7,8] , i m p r o v e d in p a r t i n C o l o m b e a u - D w y e r - P e r r o t [l]
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CHAPTER 11 T h e W e i ' e r s t r a s s p r e p a r a t i o n t h e o r e m s 11.1.3-4 a r e in M a z e t [l], 8 11.2 is taken f r o m S c h w a r t z [4]. T h e o r e m 11.3.1 w a s obtained R a m i s [l]. i n C o l o m b e a u - G a y - P e r r o t [l] P r o p o s i t i o n 11.5.2 a n d t h e o r e m 1 1 . 6 . 2 a r e in Chansolme €13.
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CHAPTER 12
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T h e o r e m s 1 2 . 6 . 1 and 1 2 . 6 . 3 w e r e obtained i n Gupta [l, 2 , 3 ] T h e o r e m s 1 2 . 7 . 1 and 1 2 . 7 . 4 w e r e obtained in C o l o m b e a u - M a t o s [3]. T h e o r e m 1 2 . 7 . 5 w a s obtained i n C o l o m b e a u - P e r r o t [6], t h e o r e m 1 2 . 8 . 2 i n BolandDineen [3] a n d t h e o r e m 1 2 . 8 . 3 i n C o l o m b e a u - P a q u e s [3)
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P a r t i c u l a r c a s e s of t h e o r e m s 12.7.1, 1 2 . 7 . 4 a n d 1 2 . 7 . 5 a r e i n Matos [5], B e r n e r [l) , Boland [2, 39 A l r e a d y i n 1970 M a t o s e x t e n d e d G u p t a ' s r e s u l t s [l, 2 , 31 to l o c a l l y convex s p a c e s i n M a t o s [3, 41 T h e n Boland [2, 31 obtained r e s u l t s i n n u c l e a r s p a c e s . F o r c o n n e c t i o n s between t h e s e l a s t r e s u l t s s e e C o l o m b e a u - M a t o s [l] . L e t u s a l s o m e n t i o n Matos [ 6 , 8 ] , a s u r v e y a n d f u r t h e r r e f e r e n c e s in C o l o m b e a u - M a t o s [ 2 ) . Convolution e q u a t i o n s i n s p a c e s of h o l o m o r p h i c g e r m s a r e s t u d i e d i n Biagioni[l]
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CHAPTER 13 T h e o r e m 1 3 . 4 . 1 i s i n C o l o m b e a u - P a q u e s [3]. T h e s e r e s u l t s , a s well
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a s o t h e r r e s u l t s , w e r e obtained independently with a d i f f e r e n t proof by S c h w e r d t f e g e r [l] . O t h e r r e s u l t s a r e i n A n s e m i l - P e r r o t [l]
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CHAPTER 14 $1 r e v i e w s c l a s s i c a l definitions on pseudo-convexity i n l o c a l l y convex s p a c e s . T h i s m a t e r i a l i s given i n m u c h m o r e d e t a i l i n N o v e r r a z [l, 43 T h e o r e m 1 4 . 2 . 3 i s in G r u m a n - K i s e l m a n [l] , t h e o r e m 1 4 . 2 . 5 i s in Colombeau-Mujica [3] but follows a l s o e a s i l y f r o m S c h o t t e n l o h e r €43 . T h e o r e m 1 4 . 3 . 8 i s t a k e n f r o m N o v e r r a z [3,4] a n d t h e o r e m 14.4.1 f r o m C o l o m b e a u - P e r r o t [ll] w h e r e it is a l e m m a f o r the s o l u t i o n of t h e 6 e q u a t i o n
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T h e L k v i p r o b l e m w a s s o l v e d in s e p a r a b l e H i l b e r t s p a c e s a n d i n l i n e a r s p a c e s equipped with t h e finite d i m e n s i o n a l bornology by G r u m a n [l], t h e n i n B a n a c h s p a c e s with b a s i s i n G r u m a n - K i s e l m a n [l]. A c o u n t e r e x a m p l e i n non s e p a r a b l e B a n a c h s p a c e s is in Josefson[3].The L k v i p r o b l e m w a s t h e n s o l v e d in Silva s p a c e s with b a s i s i n Pome's [l]. V e r y g e n e r a l r e s u l t s w e r e obtained in Schottenloher [4] f o r d o m a i n s s p r e a d o v e r locally convex s p a c e s with a S c h a u d e r d e c o m p o s i t i o n . A f t e r t h e s e l a s t r e s u l t s w e r e obtained, o t h e r p r o o f s i n F r C c h e t s p a c e s with b a s i s a n d Silva s p a c e s with b a s i s w e r e published i n D i n e e n - N o v e r r a z - S c h o t t e n l o h e r [l], a s w e l l a s t h e c a s e of all n u c l e a r Silva s p a c e s i n C o l o m b e a u - M u j i c a [ 3 ] , S e e a l s o Mujica [4] f o r a s u r v e y a n d new r e s u l t s . See t h e H i s t o r i c a l N o t e s i n Dineen [l] f o r t h e evolution of t h i s p r o b l e m i n infinite d i m e n s i o n . CHAPTER 15 T h e o r e m 1 5 . 7 . 1 i s i n C o l o m b e a u - P e r r o t [ll], ( s e e a l s o N o s s k e equa[l]). A w e a k e r r e s u l t w a s i n Raboin [7] a n d t h e r e s o l u t i o n of t h e t i o n i n t h e whole s p a c e w a s in C o l o m b e a u - P e r r o t [lo] T h e o r e m 1 5 . 8 . l is in C o l o m b e a u - M u j i c a [l].
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T h e 5 equation i n s e p a r a b l e H i l b e r t s p a c e s w a s s t u d i e d by H e n r i c h [l] when t h e s e c o n d m e m b e r h a s a polynomial g r o w t h . He u s e s i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e i n H i l b e r t s p a c e s a n d h e o b t a i n s solutions defined on a d e n s e s u b s p a c e . L a t e r Raboin el, 2, 3 , 4 , 5, 6,7] s t u d i e d t h e b equation i n a r b i t r a r y pseudo-convex open s u b s e t s of s e p a 2 r a b l e H i l b e r t s p a c e s , without g r o w t h condition. He u s e s H ( 3 r m a n d e r ' s L e s t i m a t e s a n d i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e . When t h e s e c o n d m e m b e r i s Coo a n d of bounded t y p e , h e o b t a i n e d e x i s t e n c e of C 1 solutions defined o n a d e n s e s u b s p a c e . H i s t h e o r e m is published in Raboin [l, 2, 3, 4, 5, 6, 71 i n which i t s proof i s s k e t c h e d at v a r i o u s l e v e l s . It is v e r y c l o s e t o a n i m p r o v e m e n t of the l e m m a 1 5 . 4 . 1 of t h i s book. I n p a r t i c u l a r he obtained a s c o n s e q u e n c e s s o m e e x i s t e n c e r e s u l t in n u c l e a r Silva s p a c e s roblem in these with b a s i s (Raboin [7] ) a n d t h e solution of t h e f i r s t C o u s i n ps p a c e s (Raboin [6, 71). T h e n i n C o l o m b e a u - P e r r o t [lo] the 0 e q u a t i o n w a s s o l v e d , i n the c a s e of all n u c l e a r Silva s p a c e s , but only i n the whole s p a c e . F o r t h i s a c o m p l i c a t e d i m p r o v e m e n t of t h e a b o v e R a b o i n ' s proof w a s u s e d t o obtain i n t h e H i l b e r t i a n f r a m e w o r k a Cw solution on a d e n s e s u b s p a c e , which w a s u s e d a s a l e m m a . T h e n t h i s r e s u l t w a s extended t o a r b i t r a r y pseudo-convex open s u b s e t s of n u c l e a r Silva s p a c e s i n C o l o m b e a u -
430
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P e r r o t [ll], and independently in N o s s k e el]. L a t e r a l a r g e p a r t of R a b o i n ' s proof a n d all i t s long i m p r o v e m e n t i n C o l o m b e a u - P e r r o t [lo] w e r e r e p l a c e d by a c o n s i d e r a b l y s h o r t e r proof of a n hypoellipticity r e s u l t due t o Mazet [2] ( l e m m a 15.5.1 of t h i s book). Still l a t e r a t e c h n i c a l a s s u m p t i o n on the s e c o n d m e m b e r , c o n s i d e r e d i n Raboin [7], w a s p r o v e d t o be a l w a y s t r u e i n Colombeau-Mujica [l] ( t h e o r e m 1 . 6 . 3 of t h i s book), S O t h a t t h e conjunction of Raboin [ 7 ] , Mazet [2] a n d Colombeau-Mujica [l] g i v e s in t h e p a r t i c u l a r c a s e of n u c l e a r Silva s p a c e s with b a s i s a proof d i f f e r e n t f r o m the f o r m e r proofs i n C o l o m b e a u - P e r r o t [ll] a n d N o s s k e [l] (that do not u s e the b a s i s a s s u m p t i o n ) . T h i s c o n c e r n e d the c a s e of 0 , l f o r m s ; the c a s e of p , q f o r m s with q > 1 r e m a i n s unsolved. V e r y i n t e r e s ting c o u n t e r e x a m p l e s of a different n a t u r e , i n Dineen [ 8 ] and M e i s e Vogt [3], show that even t h e c a s e of 0 , l f o r m s is not in g e n e r a l s o l vable in n u c l e a r F r k c h e t s p a c e s . F i n a l l y we m u s t mention that a " p e r sonal" a p p r e c i a t i o n on the r e s p e c t i v e contributions of s o m e of t h e s e a u t h o r s was published in K r a m m [l]. CHAPTER 16 T h e o r e m 16.1.1 is a s t a n d a r d consequence of t h e o r e m 15.7.1 a n d was f o r m e r l y obtained, a s a l r e a d y quoted a b o v e , i n t h e p a r t i c u l a r c a s e of s p a c e s with b a s i s i n Raboin [6, 74. T h e c o u n t e r e x a m p l e i n $ 16.2 i s taken f r o m Dineen [ 8 ] . T h e o r e m 16.3.1 w a s published i n C o l o m b e a u G a y - P e r r o t [l] . O t h e r applications a r e in K r a m m
111.
F i r s t we quote t h e s o u r c e of t h e t h e o r e m s p r e s e n t e d i n t h e book, a n d t h e n we give s o m e m o r e r e f e r e n c e s of p r e v i o u s o r r e l a t e d r e s u l t s . T h e r e f o r e n u m e r o u s i m p o r t a n t p a p e r s c o n n e c t e d with t h e t o p i c s d i s c u s s e d a r e not m e n t i o n e d . CHAPTER 1 T h e o r e m s 1 . 4 .7-8 w e r e obtained i n C o l o m b e a u [3,17] a n d t h e o r e m s 1 . 6 . 3 w a s obtained in C o l - m b e a u - M u j i c a [l] . T h e concept of differentiable m a p p i n g s in t h e s e n s e of 8 1.1 a n d 6 1 . 2 i s i n Sebastigo e Silva [l, 21 in t h e c a s e of t h e Von N e u m a n n b o r nologies of l o c a l l y convex s p a c e s . T h e n h e p l a c e d h i s t h e o r y i n i t s n a t u r a l s e t t i n g of bornological v e c t o r s p a c e s in S e b a s t i c o e Silva [3]. T h e c o n Cco m a p p i n g s in the e n l a r g e d s e n s e c o n s i d e r e d in $ 1 . 4 w a s i n c e p t of t r o d u c e d in Sebastia40 e Silva [ Z ] . T h e concept of d i f f e r e n t i a b l e m a p p i n g s in t h e s e n s e of $ 1.1 a n d 8 1 . 2 w a s a l s o i n t r o d u c e d i n M a r i n e s c u [l] in the s e t t i n g of bornological v e c t o r s p a c e s ( p o l y n o r m e d s p a c e s i n M a r i n e s c u [ l ] ' s t e r m i n o l o g y ) . T h e r e i s a slight d i f f e r e n c e b e t w e e n M a r i n e s c u ' s definition a n d t h e definition in $ 1.1, 0 1 . 2 , s e e C o l o m b e a u [ 2 ] , p . 20 a n d p.31-32. F o r f u r t h e r c o m p a r i s o n r e s u l t s between t h e t h r e e c o n c e p t s of differentiability of 3 1 . 1 , 0 1 . 2 , 8 1 . 4 a n d $ 1. 5 we r e f e r to C o l o m b e a u [ 3 , 4,171 . F o r the g e n e r a l l i t t e r a t u r e on d i f f e r e n t i a b l e m a p p i n g s between l o c a l l y convex s p a c e s we r e f e r t o t h e e x c e l l e n t s u r v e y s of A v e r b u c k Smolyanov [ Z ] a n d N a s h e d [l] t h a t contain e x c e l l e n t b i b l i o g r a p h i e s f o r p a p e r s a n t e r i o r to 1969. Among t h e v e r y n u m e r o u s books a n d a r t i c l e s on t h i s s u j e c t l e t u s f u r t h e r m o r e quote Averbuck-Smolyanov [l] , B u c h e r F r b l i c h e r [l] , K e l l e r [l] , Y a m a m u r o [l, 21, M e i s e [l], C o l o m b e a u Meise [l]. In t h i s book we did n o t c o n s i d e r t h e v e r y i m p o r t a n t t o p i c s of I m p l i c i t F u n c t i o n s a n d O r d i n a r y D i f f e r e n t i a l E q u a t i o n s . In t h e c a s e of Banach s p a c e s v e r y g e n e r a l r e s u l t s w e r e obtained a l r e a d y i n t h e t h i r t i e s ( s e e C a r t a n [l] o r DieudonnC [l] f o r i n s t a n c e ) a n d a r e now v e r y c l a s s i c a l . In the c a s e of non n o r m e d s p a c e s t h e s i t u a t i o n i s c o n s i d e r a b l y m o r e c o m p l i c a t e d a n d does not s e e m t o be c o m p l e t e l y c l a r i f i e d at p r e s e n t . So we j u s t s k e t c h h e r e t h i s topic a n d give s o m e r e f e r e n c e s . L e t u s f i r s t a s s u m e the e x i s t e n c e of a continuous i m p l i c i t f u n c tion o r of a continuous i n v e r s e , when t h e g i v e n f u n c t i o n s a r e d i f f e r e n t i a b l e In t h e Banach s p a c e c a s e t h i s i m p l i c i t function o r t h i s i n v e r s e m a p i s diff e r e n t i a b l e (Nachbin [43, P r o p o s i t i o n 16.16 a n d 22.10 f o r i n s t a n c e ) . A c o u n t e r e x a m p l e i n Averbuck-Smolyanov [ 2 ) shows that t h i s is no l o n g e r t r u e in g e n e r a l l o c a l l y convex s p a c e s . On t h i s topic s e e C o l o m b e a u [ 4 ] , Smolyanov [l]. If one a s s u m e s s o m e p r o p e r t y s t r o n g e r t h a n continuity on t h e i m p l i c i t o r i n v e r s e m a p , one o b t a i n s the d e s i r e d differentiability 423
Bibliographic Notes
424
r e s u l t , s e e C o l o m b e a u [4], t h e o r e m 2 . 2 which i s q u i t e g e n e r a l . Now l e t i s a n - t i m e s d i f f e r e n t i a b l e b i j e c t i o n a n d t h a t its i n v e r s e m a p us assume f f - l i s one t i m e d i f f e r e n t i a b l e ( s a m e s i t u a t i o n a n d r e s u l t s i n t h e c a s e of t h e i m p l i c i t f u n c t i o n ) . In t h e c a s e of B a n a c h s p a c e s f - l i s n - t i m e s d i f f e rentiable. A counterexample in Colombeau [4], $ 6 , example 2 , shows t h a t t h i s i s no l o n g e r t r u e i n g e n e r a l i n non n o r m e d s p a c e s , but t h e o r e m 3 . 2 i n C o l o m b e a u [4] g i v e s a v e r y g e n e r a l c a s e i n which f - l i s n - t i m e s d i f f e r e n t i a b l e . In s h o r t t h e r e s u l t s a r e q u i t e g e n e r a l , but one n e e d s t o a s s u m e m o r e r e s t r i c t i v e a s s u m p t i o n s t h a n in t h e B a n a c h s p a c e s c a s e (the continuity of t h e i m p l i c i t f u n c t i o n i m p l i e s t h e s e p r o p e r t i e s i n t h e B a n a c h s p a c e s c a s e ) . The s i t u a t i o n of t h e e x i s t e n c e r e s u l t s i s m u c h w o r s e : L e t u s now c o n s i d e r t h e p r o b l e m of e x i s t e n c e of a n i m p l i c i t f u n c t i o n o r of a n i n v e r s e of a g i v e n d i f f e r e n t i a b l e m a p , in t h e c o n d i t i o n s t h a t , i n B a n a c h s p a c e s , e n s u r e t h e i r e x i s t e n c e ( C a r t a n [ l ] , Dieudonng [l], . ) . C o u n t e r e x a m p l e s i n E e l l s [l] , P i s a n e l l i [l] , C o l o m b e a u [l] , L o j a s i e w i c z J r . [l] show t h a t t h e v e r y n e a t r e s u l t s of t h e B a n a c h s p a c e s f r a m e w o r k do not r e m a i n v a l i d i n l o c a l l y c o n v e x s p a c e s . T h e r e is a v e r y l a r g e a m o u n t of w o r k s on t h i s t o p i c : people f i r s t t r i e d v a r i o u s g e n e r a l i z a t i o n s of the c l a s s i c a l p r o o f , s e e f o r i n s t a n c e F a l b - J a c o b [l], M a c D e r m o t t [l, 2 , 31, C o l o m b e a u [l, 5, 6 , 7 1 , Y a m a m u r o [ 2 ] , T h e i m p l i c i t function a n d l o c a l i n v e r s i o n t h e o r e m s i n C o l o m b e a u [l, 5, 61 a r e u s e d i n J . A . L e s l i e [ 3 ] f o r a proof of a K u p k a - S m a l e t h e o r e m in t h e r e a l a n a l y t i c c a s e . A d i f f e r e n t m e t h o d w a s i n s p i r e d by t h e " N a s h I m p l i c i t F u n c t i o n T h e o r e m " , s e e J . T . S c h w a r z [l], M o s e r [l] , H a m i l t o n [l] , S e r g e r a e r t [l] , J a c o b o w i t z [ l ] , Z e h n d e r [l], L o j a s i e w i c z - Z e h n d e r [l] T h i s o t h e r kind of i m p l i c i t function t h e o r e m s have c l a s s i c a l p o w e r f u l a p p l i c a t i o n s , f o r i n s t a n c e in t h e proof of t h e e m b e d d i n g of R i e m a n n i a n m a n i f o l d s i n IR" , s e e S c h w a r z [l]
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Now l e t us c o n s i d e r " o r d i n a r y " d i f f e r e n t i a l e q u a t i o n s X ' ( t ) = F ( X ( t ) ,t ) , X(to) = Xo ( C a u c h y p r o b l e m s ) . T h e v e r y b e a u t i f u l e x i s t e n c e , u n i q u e n e s s a n d d e p e n d e n c e on p a r a ' m e t e r s a n d d a t a r e s u l t s of t h e B a n a c h s p a c e c a s e t h e o r y ( C a r t a n [l] , Dieudonne [l] , . ) do not r e m a i n v a l i d i n l o c a l l y convex s p a c e s , s e e C o l o m b e a u [2] , P l i s [l], De G i o r g i [l]). V a r i o u s m e t h o d s w e r e u s e d : a c o m p a c t n e s s m e t h o d in Dubinsky [l], a n , i t e r a t i o n m e t h o d in C o l o m b e a u 61, 2 , 61, T r e v e s [2] , L e s l i e [2], V e r y good r e s u l t s a n d a p p l i c a t i o n s c a m e f r o m a v e r y s t r o n g r e i n f o r c e m e n t of t h i s i t e r a t i o n m e t h o d by a n a s t u t e m a j o r i z a t i o n t e c h n i q u e i n t h e s o c a l l e d "Ovcyannikov method" : Ovcyannikov [l, 2 3 , T r e v e s [ 3 , 4 , 51, S t e i m b e r g T r e v e s [l], S t e i m b e r g [l], P i s a n e l l i [2], N i r e m b e r g [l], Du C h a t e a u [l], L a s c a r [2], . T h e s e r e f e r e n c e s c o n c e r n t h e s t u d y at o r d i n a r y points (for a c l a s s i f i c a t i o n of s i n g u l a r i t i e s of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s s e e Wasow [ l ] ) . T h e Ovcyannikov m e t h o d w a s u s e d i n t h e c a s e of " r e g u l a r s i n g u l a r points" i n Baouendi-Goulaouic [l, 21 a n d i n t h e c a s e of " i r r e g u l a r s i n g u l a r points" i n C o l o m b e a u - M C r i l [l]
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B e s i d e s the a b o v e t o p i c s , D i f f e r e n t i a l C a l c u l u s i n a . c . s . h a s a l o t of v a r i o u s a p p l i c a t i o n s , m o s t of t h e m being at p r e s e n t i n f u l l d e v e l o p -
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Bibliographic Notes
m e n t : f o r i n s t a n c e d i f f e r e n t i a l s t r u c t u r e s i n s e t s of COD o r real a n a l y tic d i f f e o m o r p h i s m s of c o m p a c t R i e m a n n i a n m a n i f o l d s ( L e s l i e [l, 31 ) o r new c o n c e p t s of g e n e r a l i z e d f u n c t i o n s on IE?? , giving a m e a n i n g t o a n y p r o d u c t of d i s t r i b u t i o n s ( C o l o m b e a u [l8] ), j u s t t o quote a few of t h e m . CHAPTER 2 T h e o r e m 2.2.3 w a s obtained in L a z e t [l, 2 3 . T h e o r e m s 2 . 3 . 3 - 4 a r e i n C o l o m b e a u [ 8 ] . T h e o r e m 2 . 4 . 1 and c o r o l l a r i e s w e r e o b t a i n e d i n L a z e t [l, 21, C o l o m b e a u [9,10] T h e c o u n t e r e x a m p l e in 2 . 5 w a s obtained i n C o l o m b e a u [9,10] . T h e o r e m 2 . 6 . 4 w a s o b t a i n e d i n B o c h n a k - S i c i a k [ l , 23 a n d T h e o r e m 2 . 7 . 4 i n C o l o m b e a u - M u j i c a [l]
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T h e concept of Silva h o l o m o r p h i c m a p p i n g s w a s i n t r o d u c e d in S e b a s t i s e Silva [l, 2 , 33, a n d t h e n s t u d i e d i n m o r e d e t a i l s in C o l o m b e a u [ 9 , 1 0 , 8 , 1 , 1 7 ] , C o l o m b e a u - L a z e t [l], L a z e t [l, 23, P i s a n e l l i [3, 43, M a t o s Nachbin [l] , B i a n c h i n i [l] , B i a n c h i n i - P a q u e s - Z a i n e [l], e t c . L e t u s m e n tion a n i c e o r i g i n a l i n t r o d u c t i o n i n M e i s e - V o g t [l] a n d t h a t s e v e r a l a u t h o r s i n t r o d u c e d t h e concept of "hypo-analytic m a p p i n g s " which c o i n c i d e s with t h e one of Silva h o l o m o r p h i c m a p p i n g s i n all "usual" A . c s . .
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T h e c o n c e p t of h o l o m o r p h i c ( = G - a n a l y t i c a n d c o n t i n u o u s ) m a p p i n g s i n l o c a l l y convex s p a c e s h a s b e e n s t u d i e d by a c o n s i d e r a b l e n u m b e r of a u t h o r s , a n d we r e f e r t o D i n e e n ' s r e c e n t book [I] f o r r e f e r e n c e s , a s w e l l a s t o Nachbin [l] f o r a n i n t r o d u c t i o n a n d r e f e r e n c e s . T h e r e a r e a l s o m a n y books a n d v o l u m e s of P r o c e e d i n g s c o n c e r n i n g t h i s s u b j e c t : B a r r o s o [l, 2 , 3, 41, Boland [3], C o e u r 6 [l], Dineen [l] , Hayden-Suffridge [l] , HervC [l], Lelong [ Z , 33, Lelong-Skoda [l], Machado [l] , M a t o s [7], Mazet [l] , Mujica [l] , Nachbin [ Z ] , N o v e r r a z [l, 43, Z a p a t a [l] , e t c . L e t u s a l s o m e n t i o n the s u r v e y a r t i c l e s N a c h b i n [l, 3 , 51, B i e r s t e d t - M e i s e [Z] , C o l o m b e a u - M a t o s [ Z ] , Dineen [lo] Concerning holomorphic functions on n u c l e a r F r k c h e t s p a c e s a n d v e r y i m p o r t a n t c o u n t e r e x a m p l e s s e e M e i s e Vogt [ 2 , 31
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T h e h o l o m o r p h i c r e p r e s e n t a t i o n of F o c k s p a c e s of B o s o n f i e l d s i s e x p o s e d o r u s e d i n B e r e z i n [l] , Novozhilov-Tulub [l], S a r a v a s t i - V a l a t i n [l], R z e w u s k i [l, 2 3 , Dwyer [l, 2 , 3 , 4 J , C o l o m b e a u - P e r r o t [l, 2 , 31, K r 6 e [l, 21, K r g e - R a c z k a [l]
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CHAPTER 3 T h e o r e m s 3 . 1 . 2 , 3 . 2 . 1 , 3 . 2 . 3 a n d 3 . 3 . 1 a r e r e f o r m u l a t i o n s of c l a s s i c a l r e s u l t s . E x a m p l e 3 ' 2 . 4 is t a k e n f r o m C o l o m b e a u [9,10] ; a s i milar c o u n t e r e x a m p l e w a s o b t a i n e d independently in H i r s c h o w i t z [l] E x a m p l e 3 . 3 . 3 i s t a k e n f r o m Colombeau [17] A m o r e g e n e r a l f o r m of t h e o r e m 3 . 4 . 3 - 4 i s in C o l o m b e a u - L a z e t [l] , C o l o m b e a u [lo]
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S o m e l i t t e r a t u r e on Z o r n , H a r t o g s a n d M o n t e l ' s t h e o r e m s i s i n Dineen [ l ] , S e b a s t i c o e Silva [l, 21, Hille [l] , H i l l e - P h i l l i p s [l],
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Bibliographic Notes
L a z e t [l, 2 3 , Colombeau [l, 9,10,17], Col.ombeau-Lazet [l] , N o v e r r a z [l, 2 1 , B o c h n a k - S i c i a k [l, 21, P i s a n e l l i [3, 43, Matos [I, 23, .
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CHAPTER 4 T h e o r e m s 4 . 2 . 1 - 2 , 4 . 3 . 1 , 4 . 4 . 1 w e r e obtained i n C o l o m b e a u P e r r o t [2,4,5]. T h e o r e m s 4 . 2 . 4 , 4 . 3 . 4 , 4 . 4 . 2 w e r e obtained i n a n unpublished m a n u s c r i p t Colombeau [ll] ; s e e a l s o M e i s e [l] a n d C o l o m b e a u M e i s e [l]. T h e s t r i c t l y c o m p a c t p o r t e d topology w a s i n t r o d u c e d i n BianchiniP a q u e s - Z a i n e [l] a n d p r o p o s i t i o n 4 . 1 . 4 w a s obtained i n C o l o m b e a u - M e i s e P e r r o t [l] N a c h b i n ' s p o r t e d t o p o l o g y w a s defined i n Nachbin [2] f o r hol o m o r p h i c functions on Banach s p a c e s . T h i s last book o r i g i n a t e d a v e r y i m p o r t a n t t r e n d of w o r k s , in p a r t i c u l a r on topologies i n s p a c e s of holom o r p h i c m a p p i n g s , a n d t h i s last topic i s t r e a t e d i n d e t a i l i n t h e r e c e n t book Dineen [l], to which we r e f e r . Let u s j u s t m e n t i o n r e c e n t r e s u l t s i n Dineen [9], Boland-Dineen [4], M e i s e [ 21, Mujica €21
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CHAPTER 5 T h e o r e m 5.1.2 w a s obtained i n the m o r e g e n e r a l c a s e of a f i n i t e l y Runge open s e t 0 in C o l o m b e a u - M e i s e - P e r r o t [l]. T h e o r e m 5 . 2 . 1 w a s obtained in C o l o m b e a u - M e i s e [l], s e e a l s o M e i s e [l] T h e proof of t h e o r e m 5 . 3 . 1 is a n i m m e d i a t e a d a p t a t i o n of a proof i n B o n i c - F r a m p t o n [I].
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G e n e r a l i z a t i o n s of N a c h b i n ' s A p p r o x i m a t i o n T h e o r e m a r e i n P r o l l a [ 2 , 31, G u e r e i r o - P r o l l a [l], s e e a l s o Nachbin [7]
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T h e r e s u l t s in t h i s c h a p t e r w e r e c h o s e n s i n c e we u s e t h e m i n t h i s book. S e v e r a l o t h e r r e s u l t s of e x i s t e n c e a n d a p p r o x i m a t i o n a r e i n c h a p t e r s 6 , 9 t o 16. S o m e r e s u l t s which a r e not i n t h i s book a r e s u r v e y e d i n : Dineen [l], J o s e f s o n Schottenloher [l] T h e y c o n c e r n (1) . , ( 2 ) e x i s t e n c e of ho[ 1 , 2 3 , R u s e k [I], Bayoumi [l], l o m o r p h i c functions with p r e s c r i b e d r a d i u s of c o n v e r e n c e : A r o n T i f , C o e u r 6 [2], K i s e l m a n 11, 2 , 31, Schotten-&rvey in Schottenloher [l] contains a l s o o t h e r t o p i c s r e l a t e d with following c h a p t e r s of t h i s book. E x i s t e n c e r e s u l t s on h o l o m o r p h i c f u n c t i o n s with p r e s c r i b e d v a l u e s a t a n infinite given s e t of points a r e in H e r v i e r [I], Valdivia [l] R e s u l t s on e x t e n s i o n s of h o l o m o r p h i c f u n c t i o n s defined in a c l o s e d s u b s p a c e of a l o c a l l y convex s p a c e a r e i n B o l a n d r 3 ] , A r o n - B e r n e r [l], Colombeau-Mujica Meise-Vogt [3] R e s u l t s of e x i s t e n c e of Co3 holomorphic ma in s with p r e s c r i b e d a s y m p t o t i c e x a n s i o n s a r e i n G b e a T d k & - 5 , 1 6 ] , Colombeau-Mujici&zr.
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CHAPTER 6 T h e o r e m 6.1.1 w a s obtained in a m o r e r e s t r i c t i v e c a s e i n A r o n Schottenloher [l], Schottenloher [3], a n d in m o r e g e n e r a l i t y i n P a q u e s [l]
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C o r o l l a r y 6.1.4 w a s d i r e c t l y obtained f o r Cn f u n c t i o n s in M e i s e [l] w h e r e a d e t a i l e d proof is g i v e n . A m o r e g e n e r a l f o r m of t h e o r e m 6 . 2 . 1 i s in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 5 . C o r o l l a r y 6 . 2 . 2 w a s d i r e c t l y obtained in Colombeau [ll] a n d i s a l s o in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 6 T h e o r e m 6 . 3 . 2 i s i n C o l o m b e a u - M e i s e - P e r r o t [l] A m o r e g e n e r a l f o r m of t h e o r e m 6 . 3 . 3 i s in C o l o m b e a u - M e i s e [l] r e m a r k 4 . 5 .
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F o r o t h e r p a p e r s on t h e A p p r o x i m a t i o n P r o p e r t y of 'Q (Q), X(Q) a n d Cn(Q) s e e P a q u e s [2], A r o n - S c h o t t e n l o h e r [l] , M e i s e [l], B o m b a l Gordon-Gonzalez Llavona [l] . U s e of the " k e r n e l t h e o r e m " 6.1.4 t o a study of l i n e a r o p e r a t o r s o n F o c k s p a c e s of Boson f i e l d s i s done in C o l o m b e a u - P e r r o t [l, 2 , 3 ] , Kre'e [l, 21, K r i e - R a c z k a [l]
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CHAPTER 7 T h e o r e m 7 . 2 . 1 w a s obtained in C o l o m b e a u - P e r r o t [6] but is a l s o a c o n s e q u e n c e of a t h e o r e m in Boland (2, 31, s e e C o l o m b e a u - P e r r o t [6]. T h e o r e m 7 . 4 . 1 is i n A n s e m i l - C o l o m b e a u [l] a n d t h e o r e m " 7 . 4 . 7 i n C o l o m b e a u - P o n t e [l]
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T h e f i r s t r e s u l t o n t h e F o u r i e r - B o r e 1 i s o m o r p h i s m i n infinite d i m e n s i o n w a s obtained by Gupta [l, 2 , 3 ] f o r e n t i r e f u n c t i o n s of n u c l e a r type on Banach s p a c e s . T h i s r e s u l t w a s e x t e n d e d to l o c a l l y convex s p a c e s in M a t o s [ 3 , 4 ] a n d Boland [ 2 , 3 ] . F o r connections b e t w e e n t h e above r e s u l t s of Boland a n d Matos s e e C o l o m b e a u - M a t o s [l] . S p a c e s of h o l o m o r p h i c g e r m s w e r e i n t e n s i v e l y s t u d i e d , s e e t h e book Dineen [l] ; let u s only quote Mujica [l. 2, 31, Dineen [5], B i e r s t e d t M e i s e (1, 21, A r a g o n a [l] , S o r a g g i [l] , Biagioni [l]
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T h e difficulty to extend the c l a s s i c a l P a l e y - W i e n e r - S c h w a r t z t h e o r e m t o infinite d i m e n s i o n w a s pointed out i n Dineen-Nachbin [l] T h e n a P. W .S. t h e o r e m w a s obtained i n A b u a b a r a [l, 21 a n d h i s i d e a w a s a d a p t e d t o the c a s e of n u c l e a r s p a c e s i n A n s e m i l - C o l o m b e a u [l]. O t h e r r e s u l t s w e r e obtained by defining a s u i t a b l e s p a c e of Coo f u n c t i o n s i n C o l o m b e a u Ponte [l] f o r t h e c a s e of n u c l e a r s p a c e s a n d in C o l o m b e a u - P a q u e s [l] f o r the c a s e of Banach s p a c e s . See a l s o o t h e r kinds of P . W . S . t h e o r e m s in t h i s book c h a p t e r 13 ( C o l o m b e a u - P a q u e s [3] ), i n C o l o m b e a u - P a q u e s [a] a n d Gal6 [l]
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CHAPTER 8 T h e o r e m s 8.1.1-2 a r e r e f o r m u l a t i o n s of a r e s u l t obtained i n d e pendently i n Boland [4] a n d W a e l b r o e c k [I] T h e proof g i v e n h e r e is t h a t of C o l o m b e a u - P e r r o t [ 7 ] obtained l a t e r . T h e o r e m s 8 . 2 . 5 - 6 w e r e g i v e n in C o l o m b e a u - M e i s e [2] a n d a r e e x t e n s i o n s of a r e s u l t i n B i e r s t e d t - G r a m s c h Meise [l] T h e o r e m 8 . 3 . 2 i s due to M e i s e [ 3 ] , s e e a l s o C o l o m b e a u - M e i s e
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bl
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428
Bibliographic Notes
M o r e r e c e n t l y in B o r g e n s - M e i s e - V o g t [l, 2, 31 a p p e a r e d a unified proof of all n u c l e a r i t y r e s u l t s p r e s e n t l y known in Infinite D i m e n s i o n a l Holomorphy, a n d i t is e a s y to s e e t h a t t h e m e t h o d s t h e r e a l s o apply t o the s p a c e s K s ( n ) s i n c e the p r o o f s r e l y on l e m m a s dealing with t h e B a n a c h space situation. CHAPTER 9 T h e o r e m 9 . 4 . 1 i s in C o l o m b e a u - P e r r o t [8], a s well a s a p h y s i c a l i n t e r p r e t a t i o n of t h e s e e q u a t i o n s . A d i f f e r e n t proof a n d f u r t h e r r e s u l t s a r e in Colombeau-Matos [2]. CHAPTER 10 T h e o r e m s 1 0 . 2 . 6 and 1 0 . 3 . 4 a r e i n C o l o m b e a u - P e r r o t [9] a n d we expose h e r e the proof of t h i s p a p e r s i n c e we s h a l l n e e d it a l s o i n c h a p t e r 16. Another proof will be given in c h a p t e r 11, a n d will be t a k e n f r o m C o l o m b e a u G a y - P e r r o t [l] A t h i r d proof is i n C o l o m b e a u - M a t o s [ 2 ] .
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O t h e r r e s u l t s on convolution e q u a t i o n s i n s p a c e s of e n t i r e f u n c t i o n s of exponential type a r e i n Boland-Dineen [3] ; they a r e obtained f r o m t h e f i n i t e d i m e n s i o n a l r e s u l t s by a method of t r a n s f i n i t e induction. A g r e a t a m o u n t of r e s u l t s on r e l a t e d e q u a t i o n s i s i n Dwyer [l, 2, 3, 4,5, 6, 7,8] , i m p r o v e d in p a r t i n C o l o m b e a u - D w y e r - P e r r o t [l]
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CHAPTER 11 T h e W e i ' e r s t r a s s p r e p a r a t i o n t h e o r e m s 11.1.3-4 a r e in M a z e t [l], 8 11.2 is taken f r o m S c h w a r t z [4]. T h e o r e m 11.3.1 w a s obtained R a m i s [l]. i n C o l o m b e a u - G a y - P e r r o t [l] P r o p o s i t i o n 11.5.2 a n d t h e o r e m 1 1 . 6 . 2 a r e in Chansolme €13.
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CHAPTER 12
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T h e o r e m s 1 2 . 6 . 1 and 1 2 . 6 . 3 w e r e obtained i n Gupta [l, 2 , 3 ] T h e o r e m s 1 2 . 7 . 1 and 1 2 . 7 . 4 w e r e obtained in C o l o m b e a u - M a t o s [3]. T h e o r e m 1 2 . 7 . 5 w a s obtained i n C o l o m b e a u - P e r r o t [6], t h e o r e m 1 2 . 8 . 2 i n BolandDineen [3] a n d t h e o r e m 1 2 . 8 . 3 i n C o l o m b e a u - P a q u e s [3)
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P a r t i c u l a r c a s e s of t h e o r e m s 12.7.1, 1 2 . 7 . 4 a n d 1 2 . 7 . 5 a r e i n Matos [5], B e r n e r [l) , Boland [2, 39 A l r e a d y i n 1970 M a t o s e x t e n d e d G u p t a ' s r e s u l t s [l, 2 , 31 to l o c a l l y convex s p a c e s i n M a t o s [3, 41 T h e n Boland [2, 31 obtained r e s u l t s i n n u c l e a r s p a c e s . F o r c o n n e c t i o n s between t h e s e l a s t r e s u l t s s e e C o l o m b e a u - M a t o s [l] . L e t u s a l s o m e n t i o n Matos [ 6 , 8 ] , a s u r v e y a n d f u r t h e r r e f e r e n c e s in C o l o m b e a u - M a t o s [ 2 ) . Convolution e q u a t i o n s i n s p a c e s of h o l o m o r p h i c g e r m s a r e s t u d i e d i n Biagioni[l]
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CHAPTER 13 T h e o r e m 1 3 . 4 . 1 i s i n C o l o m b e a u - P a q u e s [3]. T h e s e r e s u l t s , a s well
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Bibliographic Notes
a s o t h e r r e s u l t s , w e r e obtained independently with a d i f f e r e n t proof by S c h w e r d t f e g e r [l] . O t h e r r e s u l t s a r e i n A n s e m i l - P e r r o t [l]
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CHAPTER 14 $1 r e v i e w s c l a s s i c a l definitions on pseudo-convexity i n l o c a l l y convex s p a c e s . T h i s m a t e r i a l i s given i n m u c h m o r e d e t a i l i n N o v e r r a z [l, 43 T h e o r e m 1 4 . 2 . 3 i s in G r u m a n - K i s e l m a n [l] , t h e o r e m 1 4 . 2 . 5 i s in Colombeau-Mujica [3] but follows a l s o e a s i l y f r o m S c h o t t e n l o h e r €43 . T h e o r e m 1 4 . 3 . 8 i s t a k e n f r o m N o v e r r a z [3,4] a n d t h e o r e m 14.4.1 f r o m C o l o m b e a u - P e r r o t [ll] w h e r e it is a l e m m a f o r the s o l u t i o n of t h e 6 e q u a t i o n
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T h e L k v i p r o b l e m w a s s o l v e d in s e p a r a b l e H i l b e r t s p a c e s a n d i n l i n e a r s p a c e s equipped with t h e finite d i m e n s i o n a l bornology by G r u m a n [l], t h e n i n B a n a c h s p a c e s with b a s i s i n G r u m a n - K i s e l m a n [l]. A c o u n t e r e x a m p l e i n non s e p a r a b l e B a n a c h s p a c e s is in Josefson[3].The L k v i p r o b l e m w a s t h e n s o l v e d in Silva s p a c e s with b a s i s i n Pome's [l]. V e r y g e n e r a l r e s u l t s w e r e obtained in Schottenloher [4] f o r d o m a i n s s p r e a d o v e r locally convex s p a c e s with a S c h a u d e r d e c o m p o s i t i o n . A f t e r t h e s e l a s t r e s u l t s w e r e obtained, o t h e r p r o o f s i n F r C c h e t s p a c e s with b a s i s a n d Silva s p a c e s with b a s i s w e r e published i n D i n e e n - N o v e r r a z - S c h o t t e n l o h e r [l], a s w e l l a s t h e c a s e of all n u c l e a r Silva s p a c e s i n C o l o m b e a u - M u j i c a [ 3 ] , S e e a l s o Mujica [4] f o r a s u r v e y a n d new r e s u l t s . See t h e H i s t o r i c a l N o t e s i n Dineen [l] f o r t h e evolution of t h i s p r o b l e m i n infinite d i m e n s i o n . CHAPTER 15 T h e o r e m 1 5 . 7 . 1 i s i n C o l o m b e a u - P e r r o t [ll], ( s e e a l s o N o s s k e equa[l]). A w e a k e r r e s u l t w a s i n Raboin [7] a n d t h e r e s o l u t i o n of t h e t i o n i n t h e whole s p a c e w a s in C o l o m b e a u - P e r r o t [lo] T h e o r e m 1 5 . 8 . l is in C o l o m b e a u - M u j i c a [l].
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T h e 5 equation i n s e p a r a b l e H i l b e r t s p a c e s w a s s t u d i e d by H e n r i c h [l] when t h e s e c o n d m e m b e r h a s a polynomial g r o w t h . He u s e s i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e i n H i l b e r t s p a c e s a n d h e o b t a i n s solutions defined on a d e n s e s u b s p a c e . L a t e r Raboin el, 2, 3 , 4 , 5, 6,7] s t u d i e d t h e b equation i n a r b i t r a r y pseudo-convex open s u b s e t s of s e p a 2 r a b l e H i l b e r t s p a c e s , without g r o w t h condition. He u s e s H ( 3 r m a n d e r ' s L e s t i m a t e s a n d i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e . When t h e s e c o n d m e m b e r i s Coo a n d of bounded t y p e , h e o b t a i n e d e x i s t e n c e of C 1 solutions defined o n a d e n s e s u b s p a c e . H i s t h e o r e m is published in Raboin [l, 2, 3, 4, 5, 6, 71 i n which i t s proof i s s k e t c h e d at v a r i o u s l e v e l s . It is v e r y c l o s e t o a n i m p r o v e m e n t of the l e m m a 1 5 . 4 . 1 of t h i s book. I n p a r t i c u l a r he obtained a s c o n s e q u e n c e s s o m e e x i s t e n c e r e s u l t in n u c l e a r Silva s p a c e s roblem in these with b a s i s (Raboin [7] ) a n d t h e solution of t h e f i r s t C o u s i n ps p a c e s (Raboin [6, 71). T h e n i n C o l o m b e a u - P e r r o t [lo] the 0 e q u a t i o n w a s s o l v e d , i n the c a s e of all n u c l e a r Silva s p a c e s , but only i n the whole s p a c e . F o r t h i s a c o m p l i c a t e d i m p r o v e m e n t of t h e a b o v e R a b o i n ' s proof w a s u s e d t o obtain i n t h e H i l b e r t i a n f r a m e w o r k a Cw solution on a d e n s e s u b s p a c e , which w a s u s e d a s a l e m m a . T h e n t h i s r e s u l t w a s extended t o a r b i t r a r y pseudo-convex open s u b s e t s of n u c l e a r Silva s p a c e s i n C o l o m b e a u -
430
Bibliographic Notes
P e r r o t [ll], and independently in N o s s k e el]. L a t e r a l a r g e p a r t of R a b o i n ' s proof a n d all i t s long i m p r o v e m e n t i n C o l o m b e a u - P e r r o t [lo] w e r e r e p l a c e d by a c o n s i d e r a b l y s h o r t e r proof of a n hypoellipticity r e s u l t due t o Mazet [2] ( l e m m a 15.5.1 of t h i s book). Still l a t e r a t e c h n i c a l a s s u m p t i o n on the s e c o n d m e m b e r , c o n s i d e r e d i n Raboin [7], w a s p r o v e d t o be a l w a y s t r u e i n Colombeau-Mujica [l] ( t h e o r e m 1 . 6 . 3 of t h i s book), S O t h a t t h e conjunction of Raboin [ 7 ] , Mazet [2] a n d Colombeau-Mujica [l] g i v e s in t h e p a r t i c u l a r c a s e of n u c l e a r Silva s p a c e s with b a s i s a proof d i f f e r e n t f r o m the f o r m e r proofs i n C o l o m b e a u - P e r r o t [ll] a n d N o s s k e [l] (that do not u s e the b a s i s a s s u m p t i o n ) . T h i s c o n c e r n e d the c a s e of 0 , l f o r m s ; the c a s e of p , q f o r m s with q > 1 r e m a i n s unsolved. V e r y i n t e r e s ting c o u n t e r e x a m p l e s of a different n a t u r e , i n Dineen [ 8 ] and M e i s e Vogt [3], show that even t h e c a s e of 0 , l f o r m s is not in g e n e r a l s o l vable in n u c l e a r F r k c h e t s p a c e s . F i n a l l y we m u s t mention that a " p e r sonal" a p p r e c i a t i o n on the r e s p e c t i v e contributions of s o m e of t h e s e a u t h o r s was published in K r a m m [l]. CHAPTER 16 T h e o r e m 16.1.1 is a s t a n d a r d consequence of t h e o r e m 15.7.1 a n d was f o r m e r l y obtained, a s a l r e a d y quoted a b o v e , i n t h e p a r t i c u l a r c a s e of s p a c e s with b a s i s i n Raboin [6, 74. T h e c o u n t e r e x a m p l e i n $ 16.2 i s taken f r o m Dineen [ 8 ] . T h e o r e m 16.3.1 w a s published i n C o l o m b e a u G a y - P e r r o t [l] . O t h e r applications a r e in K r a m m
111.