## Linear Operators: General theory |

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Page 77

( 1 ) Emämn

; M ( 3 ) sup En 1 ( amn - hm , n + 1 ) ] < 0 . M m = 0 The transformation preserves

sums of series ( i.e. mo ( no Amnan ) = noan ) if and only if the equation ( 1 ) Σηλη

...

( 1 ) Emämn

**converges**for every n ; ( 2 ) Enamn - am.n + 1**converges**for each m; M ( 3 ) sup En 1 ( amn - hm , n + 1 ) ] < 0 . M m = 0 The transformation preserves

sums of series ( i.e. mo ( no Amnan ) = noan ) if and only if the equation ( 1 ) Σηλη

...

Page 145

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

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

n }

...

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

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

n }

**converges**uniformly on S- E. The sequence In }**converges**u - uniformly to the...

Page 281

Then { fn }

subsequence of { { n }

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

fn ( 80 ) ...

Then { fn }

**converges**to to at every point of S if and only if { In } 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 ( 80 ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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Acad 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 semi-group 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