## Linear Operators: General theory |

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Page 77

( 1 ) Emamm

( 1 ) Emamm

**converges**for every n ; ( 2 ) Enlamn - m , n + 1 /**converges**for each m ; M ( 3 ) sup En 1E ( Amn - Am , n + 1 ) / < 0 . M m = 0 n = 0 The transformation preserves sums of series ( i.e. meo ( Encoamnan ) = - , an ) if and ...Page 145

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

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

**converges**u - uniformly if for each ε > 0 there is a set E € such that v ( u , E ) < ε and such that { In }**converges**uniformly on S- E. The sequence { In }**converges**u ...Page 281

Then { In }

Then { In }

**converges**to to at every point of S if and only if { { n } and every subsequence of { { n }**converges**to to quasi - uniformly on A. Proof . ... To prove the sufficiency , suppose that In ( s ) does not**converge**to fo ( $ ) .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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