## Linear Operators, Part 1 |

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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 ( x ) €

Y with f ( an ) = g ( x ) . To verify that g ( x ) depends only upon x , and not upon ...

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 ( x ) €

Y with f ( an ) = g ( x ) . To verify that g ( x ) depends only upon x , and not upon ...

Page 145

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

n } converges

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 Ee such that v ( u , E ) < ε and such that { {n } converges

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

Page 360

which vanishes in a neighborhood of a point p the sequence ( S - 1 ) ( x )

converges to zero

S6 Enlæ , x ) dz 5 M , then the convergence of Snt for a given c . o . n . system is

localized ...

which vanishes in a neighborhood of a point p the sequence ( S - 1 ) ( x )

converges to zero

**uniformly**for æ in some neighborhood of p . 21 Show that if |S6 Enlæ , x ) dz 5 M , then the convergence of Snt for a given c . o . n . system is

localized ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

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