## Linear Operators: General theory |

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Page 24

If x e X , there is a sequence of points an € A with an → X. Since { an } is a

Cauchy sequence , and f is

a Cauchy sequence . Since Y is complete , there is a point g ( x ) € Y with f ( an )

→ g ( x ) ...

If x e X , there is a sequence of points an € A with an → X. Since { an } is a

Cauchy sequence , and f is

**uniformly**continuous , the sequence { f ( ar ) } is alsoa Cauchy sequence . Since Y is complete , there is a point g ( x ) € Y with f ( an )

→ g ( x ) ...

Page 145

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

n } converges

...

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 314

A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists

a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 H ( E )

PROOF . Let K C ba ( S , E ) be weakly sequentially compact and let V = UT be ...

A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists

a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 H ( E )

**uniformly**for a € K.PROOF . Let K C ba ( S , E ) be weakly sequentially compact and let V = UT be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

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