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Page 42
As an aid for the remainder of the proof , we demonstrate the following statement
: Let B , C B have the properties ( a ) if x , y e By , then xy e Bi ; ( b ) if x € B1 , then
x + 0 ; then there is a homomorphism he : B → , such that h ( x ) = 1 for X e B , .
As an aid for the remainder of the proof , we demonstrate the following statement
: Let B , C B have the properties ( a ) if x , y e By , then xy e Bi ; ( b ) if x € B1 , then
x + 0 ; then there is a homomorphism he : B → , such that h ( x ) = 1 for X e B , .
Page 415
If X is a linear topological space , then ( ii ) co ( A ) = co ( A ) , ( iii ) co ( QA ) = Q
CO ( A ) , ( iv ) If co ( A ) is compact , then co ( A + B ) = co ( A ) + co ( B ) . PROOF .
The first part of statement ( i ) follows in an elementary fashion from Lemma 1 . 4 .
If X is a linear topological space , then ( ii ) co ( A ) = co ( A ) , ( iii ) co ( QA ) = Q
CO ( A ) , ( iv ) If co ( A ) is compact , then co ( A + B ) = co ( A ) + co ( B ) . PROOF .
The first part of statement ( i ) follows in an elementary fashion from Lemma 1 . 4 .
Page 447
Statement ( b ) follows from the inequality 28 ( x + 5 ( 41 + y2 ) ) 5 k ( x + ayz ) + F
( x + ayz ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality t ( x
, – y ) + 1 ( x , y ) 2 7 ( x , 0 ) = 0 . ( x , y ) = 0 . Statement ( e ) is trivial . Q . E . D . 4 ...
Statement ( b ) follows from the inequality 28 ( x + 5 ( 41 + y2 ) ) 5 k ( x + ayz ) + F
( x + ayz ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality t ( x
, – y ) + 1 ( x , y ) 2 7 ( x , 0 ) = 0 . ( x , y ) = 0 . Statement ( e ) is trivial . Q . E . D . 4 ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero