## Linear Operators, Part 1 |

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Page 15

( b ) For every pair of distinct points x and y , there are

and y . ( c ) For every closed set A , and every x ¢ A , there are

neighborhoods of A and x . ( d ) For every pair of

there are ...

( b ) For every pair of distinct points x and y , there are

**disjoint**neighborhoods of xand y . ( c ) For every closed set A , and every x ¢ A , there are

**disjoint**neighborhoods of A and x . ( d ) For every pair of

**disjoint**closed sets A and B ,there are ...

Page 320

The semi - variation of the vector valued measure u is defined by Ile | | ( E ) = sup

| IQ : ( E : ) ) , EEE , where the supremum is taken over all finite collections of

scalars with facil 1 and all partitions of E into a finite number of

A ...

The semi - variation of the vector valued measure u is defined by Ile | | ( E ) = sup

| IQ : ( E : ) ) , EEE , where the supremum is taken over all finite collections of

scalars with facil 1 and all partitions of E into a finite number of

**disjoint**sets in E .A ...

Page 461

and an arbitrary convex set is possible , provided they are

Theorem 2 . 8 ) . He also proved that a convex set K which is compact in the X *

topology of the normed linear space X , can be separated from an arbitrary

closed ...

and an arbitrary convex set is possible , provided they are

**disjoint**( compareTheorem 2 . 8 ) . He also proved that a convex set K which is compact in the X *

topology of the normed linear space X , can be separated from an arbitrary

closed ...

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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