## Linear Operators: General theory |

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Page 100

It follows immediately from Lemma 10 that every subset of a u - null set , and

every finite union of u - null sets , is a u - null set . Any statement concerning the

points of S is said to hold u - almost

almost ...

It follows immediately from Lemma 10 that every subset of a u - null set , and

every finite union of u - null sets , is a u - null set . Any statement concerning the

points of S is said to hold u - almost

**everywhere**, or , if u is understood , simplyalmost ...

Page 150

13 COROLLARY . ( a ) If ( S , E , u ) 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 , u ) 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 722

limato ( AR ( 2 ; A ) f ) ( s ) exists almost

if S is a topological space , and if , for each 2 > 0 , R ( 2 ; A ) ; is continuous if f is

continuous , then for 1 p < oo and f e L , the limit limeso ( AR ( 2 ; A ) | ) ( s ) exists

...

limato ( AR ( 2 ; A ) f ) ( s ) exists almost

**everywhere**for feLp , 1 Sp < 0 . Show thatif S is a topological space , and if , for each 2 > 0 , R ( 2 ; A ) ; is continuous if f is

continuous , then for 1 p < oo and f e L , the limit limeso ( AR ( 2 ; A ) | ) ( s ) exists

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

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