Linear Operators: General theory |
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Page 164
8 ) of the measures { ūn } to & by the same symbols . Since the extension of a non
- negative set function on E to £ , is non - negative , it follows that { ūn ( E ) } is a
bounded non - decreasing set of real numbers for each E€ Ex We define 2 ( E ) ...
8 ) of the measures { ūn } to & by the same symbols . Since the extension of a non
- negative set function on E to £ , is non - negative , it follows that { ūn ( E ) } is a
bounded non - decreasing set of real numbers for each E€ Ex We define 2 ( E ) ...
Page 179
Let ( S , & , u ) be a positive measure space , f a nonnegative u - measurable
function defined on S and 2 ( E ) = - f ( s ) u ( ds ) , E€ E . Let g be a non - negative
a - measurable function defined on S . Then fg is u - measurable , and Sag ( s ) (
ds ) ...
Let ( S , & , u ) be a positive measure space , f a nonnegative u - measurable
function defined on S and 2 ( E ) = - f ( s ) u ( ds ) , E€ E . Let g be a non - negative
a - measurable function defined on S . Then fg is u - measurable , and Sag ( s ) (
ds ) ...
Page 314
A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists
a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 M ( E ) 0 uniformly for a €
K . Proof . Let K C ba ( S , E ) be weakly sequentially compact and let V = UT be ...
A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists
a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 M ( E ) 0 uniformly for a €
K . Proof . Let K C ba ( S , E ) be weakly sequentially compact and let V = UT be ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
21 other sections not shown
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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |
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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero