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By Joao B. Prolla

ISBN-10: 0444850309

ISBN-13: 9780444850300

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Example text

E. $ E B;. T h e r e f o r e $ = 6x($) E Q , as d e s i r e d . (b) Since 6 x ( B i ) c B', Q c B'. On t h e o t h e r hand B ' i s convex, and weak*-compact by A l a o g l u ' s Theorem: hence the G ( Qc) B ' o b t a i n s . (By G ( Qw)e mean t h e weak*-closed convex b a l a n c e d h u l l o f Q). L e t f E Qo = t h e p o l a r of Q i n L. F o r e a c h x E X , by t h e Hahn-Banach Theorem t h e r e is $ E B; such t h a t $ ( f ( x ) ) = ) ' \ f ( x )1 1 . Then 6x($) E Q. Hence I If ( x ) 1 1 = I $ ( f ( x ) ) 1 = 16x($) ( f ) I 5 1.

Any family v = (vX; x E X ) such that vx is a seminorm on Ex for each x E X is called a w e i g h t of the vector fibration ( X ; ( E x ; x E x)). We shall restrict our attention to vector fibratians and vector spaces L of cross-sections satisfynq the following conditions: (1) X is compact; ( 2 ) each Ex is a normed space, whose norm we denote by t - * I It1 1 ; COMPACT 26 - OPEN TOPOLOGY ( 3 ) if L is a vector space of cross sections, each f on X. For e v e r y C(X)-submodule W c L, we h a v e The proof is entirely similar to that of Theorem 8, only it is much simpler.

X I; x E n f o r a l l y E Y. -1 (y)1 Then h i s u p p e r s e m i c o n t i n u o u s o n Y. PROOF For e a c h y E Y , t h e s e t {y} i s c l o s e d i n Y , therefore -1 -1 TI ( y ) i s compact i n X. Hence t h e r e is a n a E TI ( y ) such t h a t -1 h ( y ) = q ( a ) = s u p { q ( x ); x E n ( y ) 1, b e c a u s e q i s upper s e m i - . c o n t i n u o u s . So h is w e l l d e f i n e d from Y t o IR L e t r E IR s e t { x E X ; q(x) > r } i s c l o s e d , whence compact i n X . C a l l The it S i n c e n i s c o n t i n u o u s , n ( X r ) i s compact i n Y .

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Approximation of Vector Valued Functions by Joao B. Prolla

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