## Linear Operators: General theory |

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Page 193

**LEMMA**. Let ( R , ER , Q ) be the product of finite measure spaces ( S , E , u ) and ( T , ET , 2 ) . Let E be a g - null set in R. Then for 2 - almost all t , the set E ( t ) = { s | ( s , t ) e E } is a d - null set . PROOF .Page 410

A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K. The following

A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K. The following

**lemma**is an obvious consequence of Definition 1. 2**Lemma**. The intersection of an arbitrary family of convex subsets of the linear space X is convex.Page 697

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**LEMMA**. Let ( S , E , u ) be a positive measure space and let { T ( 1 , ... , tk ) , ty , ... tk > 0 } be a strongly measurable semi - group of operators in L ( S , E , u ) with T ( 4 , ... , tk ) lı , T ( 4 , ... , tx ) lo sl .### 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