## Linear Operators: General theory |

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Page 77

( 1 ) Emamn

m ; M ( 3 ) sup En E ( amnam , n + 1 ) | < 0 . M m = 0 ) The transformation

preserves sums of series ( i.e. _o ( no āmnan ) = mo an ) if and only if the

equation ( 1 ...

( 1 ) Emamn

**converges**for every n ; ( 2 ) Enlamn - am , n + 1**converges**for eachm ; M ( 3 ) sup En E ( amnam , n + 1 ) | < 0 . M m = 0 ) The transformation

preserves sums of series ( i.e. _o ( no āmnan ) = mo an ) if and only if the

equation ( 1 ...

Page 145

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

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

{ fn }

the ...

A sequence of functions { n } 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 ) < E and such that

{ fn }

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

Page 595

If each root 20 has finite order al20 ) , if the sequences { f4m ) ( 20 ) }

o 5 m < ( 20 ) , and if limn + 1 ( 20 ) + 0 ... will first be shown that , for each non -

zero x in X2 , the sequence { In ( T ) x }

If each root 20 has finite order al20 ) , if the sequences { f4m ) ( 20 ) }

**converge**foro 5 m < ( 20 ) , and if limn + 1 ( 20 ) + 0 ... will first be shown that , for each non -

zero x in X2 , the sequence { In ( T ) x }

**converges**to a non - zero element of X2 .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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