## Linear Operators: General theory |

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Page 77

( 1 ) Emamn

m ; ( 3 ) sup In E ( amn – am , n + 1 ) ] < 0 . M M m = 0 The transformation

preserves sums of series ( i . e . mo ( 2 - oamnan ) = mo an ) if and only if the

equation ( 1 ...

( 1 ) Emamn

**converges**for every n ; ( 2 ) Enamn – am , n + al**converges**for eachm ; ( 3 ) sup In E ( amn – am , n + 1 ) ] < 0 . M M m = 0 The transformation

preserves sums of series ( i . e . mo ( 2 - oamnan ) = mo an ) if and only if the

equation ( 1 ...

Page 145

A sequence of functions { In } defined on S with values in X

uniformly if for each ε > 0 there is a set EEE such that v ( u , E ) < € and such that {

In }

the ...

A sequence of functions { In } defined on S with values in X

**converges**u -uniformly if for each ε > 0 there is a set EEE such that v ( u , E ) < € and such that {

In }

**converges**uniformly on S— E. The sequence { In }**converges**u - uniformly tothe ...

Page 281

Then { { n }

subsequence of { { n }

11 implies that the condition is necessary . To prove the sufficiency , suppose that

fn ...

Then { { n }

**converges**to to at every point of S if and only if { fn } and everysubsequence of { { n }

**converges**to to quasi - uniformly on A . PROOF . Theorem11 implies that the condition is necessary . To prove the sufficiency , suppose that

fn ...

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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