## 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 Mon are all continuous with

un , E ) λ ( E ) = Σ 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 Mon are all continuous with

**respect**to the measure defined by v (un , E ) λ ( E ) = Σ 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 , Ug = 2 uniformly with

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

...

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

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

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

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 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 transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero