## Linear Operators: General theory |

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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

almost

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 , analmost

**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

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 afunction g in ...

Page 676

This shows that A ( T , n ) h converges almost

set in L , and , by Lemma 5 , sup | A ( T , n ) | ( h , s ) < oo almost

every h in Lm . Thus , by Theorem IV . 11 . 2 , the sequence AT , n ) h converges ...

This shows that A ( T , n ) h converges almost

**everywhere**for every h in a denseset in L , and , by Lemma 5 , sup | A ( T , n ) | ( h , s ) < oo almost

**everywhere**forevery h in Lm . Thus , by Theorem IV . 11 . 2 , the sequence AT , n ) h converges ...

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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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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