## Linear Operators, Part 1 |

### From inside the book

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Page 2

... contains every element a of A. Two sets are

A family of sets is a

#m .

... contains every element a of A. Two sets are

**disjoint**if their intersection is void .A family of sets is a

**disjoint**family if every pair of distinct sets in the family is**disjoint**. A sequence { an } of sets is a sequence of**disjoint**sets if an n am $ for n#m .

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 lu || ( E ) = sup |

EdiM ( E ; ) ] , Εε Σ . n 4 where the supremum is taken over all finite collections of

scalars with loil S1 and all partitions of E into a finite number of

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

EdiM ( E ; ) ] , Εε Σ . n 4 where the supremum is taken over all finite collections of

scalars with loil S1 and all partitions of E into a finite number of

**disjoint**sets in E.### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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 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 isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space 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