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30. REMARK. 24,§8. To see this, just consider the vector fibration Ex = E for all x E X, where E is a fixed normed space. And then take L=C(X;E). Obviously, if Wx = w(x) in E, then inf I UEWx 5 10 EXTREME FONCTIONALS Let S be a subset of a vector space E. , if a,b E S and 0 < t < 1, then ta +(l-t)b = x implies that a = b = x. When E is a topological vector space, the c l o s e d c o n v e x h u l l of S , denoted by co(S), is the closure of its convex hull. We shall be interested in this section in characlocally terizing E(S) , when S is a subset of the dual of some convex space L of functions, or more generally, of cross-sections.

41. , and ( 2 ) o f (1) X i s compact; each (2) i s a normed s p a c e , whose norm we Ex d e n o t e by Let 11 11 . L C v ( E x ; x E X) be a v e c t o r space o f tions. A representation of r :L + L cross-see- i s a l i n e a r map C(F;M) F i s a c o m p a c t H a u s d o r f f s p a c e p r o v i d e d w i t h a continuous TI : F + X s u c h t h a t , f o r a22 f E L and x E X , the o n t o map equa li ty where 11 Ilf(x) = sup 11 r(f) ( y ) I ; y E n-'({x~) I results. 42. * Under t h e c o n d i t i o n s o f D e f i n i t i o n 1 .

1. and The above vector-valued version of the Dieudonn6 Theorem can be generalized to tensor products E F of locally convex spaces E and F which are topological modules over some locally convex topological algebra A . 5 L e t A be a l o c a l l y c o n v e x t o p o l o g i c a l a l g e b r a , and l e t M and N b e two l o c a l l y c o n v e x s p a c e s w h i c h a r e t o p o l o g i c a l m o d u l e s o v e r A. T h e n M QA N i s d e f i n e d t o be t h e q u o t i e n t l o c a l l y c o n v e x s p a c e (M Q N) ID, where M Q N i s t h e t e n s o r p r o projective d u c t o f t h e v e c t o r s p a c e s M and N endowed w i t h t h e t e n s o r p r o d u c t t o p o l o g y , and D i s t h e c l o s e d l i n e a r s u b s p a c e of M Q N spanned by e l e m e n t s o f t h e f o r m (ax Q y-x 8 ay), a E A, 48 THE THEOREM O F DIEUDONNE xEM,yEN.