## Linear Operators: General theory |

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Page 150

( a ) If ( S , E , u ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost

( a ) If ( S , E , u ) is a measure space , a sequence of measurable functions convergent in measure has a subsequence which converges almost

**everywhere**. ( b ) It ( S , E , u ) is a finite measure space , an almost**everywhere**...Page 151

Since L , ( S , E , M , X ) is complete , there m , no m , n - 00 exists a ge Le such that lim in - glo = 0. Then by Theorem 3.6 and Corollary 13 ( a ) , a subsequence of { { n } converges to g almost

Since L , ( S , E , M , X ) is complete , there m , no m , n - 00 exists a ge Le such that lim in - glo = 0. Then by Theorem 3.6 and Corollary 13 ( a ) , a subsequence of { { n } converges to g almost

**everywhere**.Page 676

Thus , by Theorem IV.11.2 , the sequence A ( T , n ) h converges almost

Thus , by Theorem IV.11.2 , the sequence A ( T , n ) h converges almost

**everywhere**for every h in Lp . Since L , is dense in L , we may apply Lemma 5 and Theorem IV.11.2 again to see that the sequence AT , n ) f converges almost ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero