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Page 188
The best known example of Theorem 2 and its Corollary 6 is obtained by taking ( Si , Ei , Mi ) to be the Borel - Lebesgue measure on the real line for i = 1 , n . Then s P = S ; is n - dimensional Euclidean space , and u = MX .
The best known example of Theorem 2 and its Corollary 6 is obtained by taking ( Si , Ei , Mi ) to be the Borel - Lebesgue measure on the real line for i = 1 , n . Then s P = S ; is n - dimensional Euclidean space , and u = MX .
Page 246
The following corollary was established during the first part of the preceding proof . 7 COROLLARY . If { bz , ... , bn } is a Hamel basis for the normed linear space X then the functionals b ...
The following corollary was established during the first part of the preceding proof . 7 COROLLARY . If { bz , ... , bn } is a Hamel basis for the normed linear space X then the functionals b ...
Page 422
11 COROLLARY . Let f be a linear functional on the linear space X , and let be a total subspace of X * . Then the following statements are equivalent : ( i ) f is in l ' ; ( ii ) f is I - continuous ; ( iii ) { x \ / ( x ) = 0 } is I ...
11 COROLLARY . Let f be a linear functional on the linear space X , and let be a total subspace of X * . Then the following statements are equivalent : ( i ) f is in l ' ; ( ii ) f is I - continuous ; ( iii ) { x \ / ( x ) = 0 } is I ...
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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