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Page 150
13 COROLLARY . ( a ) If ( S , E , p ) is a measure space , a sequence of
measurable functions convergent in measure has a subsequence which
converges almost everywhere . ( b ) If ( S , E , u ) is a finite measure space , an
almost everywhere ...
13 COROLLARY . ( a ) If ( S , E , p ) is a measure space , a sequence of
measurable functions convergent in measure has a subsequence which
converges almost everywhere . ( b ) If ( S , E , u ) is a finite measure space , an
almost everywhere ...
Page 151
( Lebesgue dominated convergence theorem ) Let 1 sp < 0 , let ( S , E , u ) be a
measure space , and let { fn } be a sequence of functions in L , ( S , E , u , X )
converging almost everywhere to a function f . Suppose that there exists a
function g in ...
( Lebesgue dominated convergence theorem ) Let 1 sp < 0 , let ( S , E , u ) be a
measure space , and let { fn } be a sequence of functions in L , ( S , E , u , X )
converging almost everywhere to a function f . Suppose that there exists a
function g in ...
Page 676
2 , the sequence AT , n ) h converges almost everywhere for every h in Ly . Since
L , is dense in Ly we may apply Lemma 5 and Theorem IV . 11 . 2 again to see
that the sequence A ( T , n ) f converges almost everywhere for every f in Ly .
2 , the sequence AT , n ) h converges almost everywhere for every h in Ly . Since
L , is dense in Ly we may apply Lemma 5 and Theorem IV . 11 . 2 again to see
that the sequence A ( T , n ) f converges almost everywhere for every f in Ly .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
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Common terms and phrases
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
Bibliographic information
| Title | Linear Operators: General theory Volume 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics |
| Author | Nelson Dunford |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |