## Linear Operators: General theory |

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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 llon are all continuous with

um , E ) 2 ( E ) = 2 Εε Σ . n = 12 " 1 + v ( un , E ) ' and thus all belong to the

subspace ca ( S , E , 2 ) consisting of all 2 - continuous functions in ca ( S , E ) .

According ...

The functions llon are all continuous with

**respect**to the measure defined by v (um , E ) 2 ( E ) = 2 Εε Σ . n = 12 " 1 + v ( un , E ) ' and thus all belong to the

subspace ca ( S , E , 2 ) consisting of all 2 - continuous functions in ca ( S , E ) .

According ...

Page 341

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

continuous . ( iii ) lim , U = 1 uniformly with

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

u ...

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

**respect**to which every 2 in K iscontinuous . ( iii ) lim , U = 1 uniformly with

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

u ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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