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Page 54
A linear mapping of one F - space into another is continuous if and only if it maps bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X → Y be linear and continuous , and let B C X be bounded .
A linear mapping of one F - space into another is continuous if and only if it maps bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X → Y be linear and continuous , and let B C X be bounded .
Page 97
and to be bounded if u is bounded . The total variation v ( u ) of an additive set function u is important because it dominates u in the sense that v ( u , E ) 2 ( u ( E ) for E € £ ; the reader should test his comprehension of ...
and to be bounded if u is bounded . The total variation v ( u ) of an additive set function u is important because it dominates u in the sense that v ( u , E ) 2 ( u ( E ) for E € £ ; the reader should test his comprehension of ...
Page 345
if and only if it is bounded and thi ( s ) converges ( to fli ) ( s ) ) for each 8 in [ a , b ] and each j = 0 , 1 , ... , p . ( c ) CP is not weakly complete , and not reflexive . ( d ) A subset A Ç CP is conditionally compact if and ...
if and only if it is bounded and thi ( s ) converges ( to fli ) ( s ) ) for each 8 in [ a , b ] and each j = 0 , 1 , ... , p . ( c ) CP is not weakly complete , and not reflexive . ( d ) A subset A Ç CP is conditionally compact if and ...
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian 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