## Linear Operators, Part 1 |

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Page 195

with

Indeed , according to Tonelli ' s theorem , both these integrals are equal to the

integral off with

that ...

with

**respect**to one variable and then with**respect**to the other , or vice versa .Indeed , according to Tonelli ' s theorem , both these integrals are equal to the

integral off with

**respect**to the product measure ( and we have already remarkedthat ...

Page 306

The functions on are all continuous with

n = 1 2n 1 + olun , E ) and thus all belong to the subspace ca ( S , E , a )

consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon ...

The functions on are all continuous with

**respect**to the measure defined by E e E ,n = 1 2n 1 + olun , E ) and thus all belong to the subspace ca ( S , E , a )

consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon ...

Page 341

( ii ) There is a non - negative u in ba ( S , E ) with

continuous . ( iii ) lim , Ugd = a uniformly with

a countable field of subsets of a set S , and let & be the o - field generated by E ...

( ii ) There is a non - negative u in ba ( S , E ) with

**respect**to which every à in K iscontinuous . ( iii ) lim , Ugd = a uniformly with

**respect**to 2 € K . 20 Let < = { En } bea countable field of subsets of a set S , and let & be the o - field generated by E ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect 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