## Linear Operators: General theory |

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Page 24

Since { an } is a Cauchy sequence , and f is

f ( an ) } is also a Cauchy sequence . Since Y is complete , there is a point g ( « ) €

Y with / ( an ) → g ( x ) . To verify that g ( x ) depends only upon X , and not ...

Since { an } is a Cauchy sequence , and f is

**uniformly**continuous , the sequence {f ( an ) } is also a Cauchy sequence . Since Y is complete , there is a point g ( « ) €

Y with / ( an ) → g ( x ) . To verify that g ( x ) depends only upon X , and not ...

Page 145

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

} converges

the ...

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

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

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

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 M ( E ) 0

K . 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 M ( E ) 0

**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 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero