## 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 of f 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 of f with

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

Page 306

The functions Mon are all continuous with

un , E ) Ee E , = 2 " 1 + 0 ( Min , 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 ...

The functions Mon are all continuous with

**respect**to the measure defined by v (un , E ) Ee E , = 2 " 1 + 0 ( Min , 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 ...

Page 341

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

continuous . ( iii ) lim , Ugd = 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 à in K iscontinuous . ( iii ) lim , Ugd = uniformly with

**respect**to 2 € K . 20 Let = { 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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero