## Linear Operators: General theory |

### From inside the book

Results 1-3 of 77

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 ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

### 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