## Linear Operators: General theory |

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that is , for every a € A , the

that is , for every a € A , the

**function f**assigns an element f ( a ) € B. If f : A + B and g : B → C , then the ... The set f ( C ) is called the image of C and the set p ( D ) is called the inverse image of D. Iff : A + A and C CA ...Page 196

Suppose that ( S , E , u ) is a measure space and

Suppose that ( S , E , u ) is a measure space and

**F**is a u - measurable**function**whose values are in L ( T , ET , 2 ) , 1 sp < oo . For each s in S ,**F**( s ) is an equivalence class**of functions**, any pair**of**whose members coincide 2 ...Page 199

integral Ss ( s , t ) u ( ds ) , as a

integral Ss ( s , t ) u ( ds ) , as a

**function of**t , is equal to the element Ss F ( s ) q ( ds ) of L , ( T , ET , 2 , X ) . Proof . Let E , be partitioned into a sequence { En } of disjoint sets of finite 2 - measure .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian 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