## Linear Operators, Part 1 |

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Page 77

( 1 ) Emämn

m ; M M ( 3 ) sup In E ( amn - Om , n + u ) < 0 . M m = 0 ) The transformation

preserves sums of series ( i.e. meo ( 2n - ohmnan ) -o an ) if and only if the

equation ( 1 ...

( 1 ) Emämn

**converges**for every n ; ( 2 ) Enlamn - am , n + 1**converges**for eachm ; M M ( 3 ) sup In E ( amn - Om , n + u ) < 0 . M m = 0 ) The transformation

preserves sums of series ( i.e. meo ( 2n - ohmnan ) -o 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 E € £ such that v ( u , E ) < ε and such that {

fn }

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 E € £ such that v ( u , E ) < ε and such that {

fn }

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

Page 281

Then { In }

subsequence of { { n }

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

( s ) ...

Then { In }

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

**converges**to to quasi - uniformly on A. Proof . Theorem 11implies that the condition is necessary . To prove the sufficiency , suppose that Ín

( s ) ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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