## Linear Operators: General theory |

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**LEMMA**. If B is a family of subsets of X , and if t is the family of all unions of subfamilies of B , then t is a topology if and only if ( i ) for every pair U ...Page 193

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**LEMMA**. Let ( R , ER , 0 ) be the product of finite measure spaces ( S , E , u ) and ( T , ET , A ) . Let E be a Q - null set in R. Then for h - almost all t , the set E ( t ) = { 8 } ( s , t ) € E } is a u - null set . PROOF .Page 700

Imbit ... dito SP ( 4 , ... , Ge } dt ... de This shows that if the

Imbit ... dito SP ( 4 , ... , Ge } dt ... de This shows that if the

**lemma**is known to be true for an integer k it is also true for any integer m S k . Thus , to prove the**lemma**, it will suffice to show that it is true if k is even .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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