## Linear Operators: General theory |

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Page 19

The metric topology in X is the smallest topology containing the spheres . The set

X , with its metric topology , is called a

such that there exists a metric function whose topology is the same as the original

...

The metric topology in X is the smallest topology containing the spheres . The set

X , with its metric topology , is called a

**metric space**. If X is a topological spacesuch that there exists a metric function whose topology is the same as the original

...

Page 20

A sequence { an } in a

= 0 . If every Cauchy sequence is convergent , a

complete . The next three lemmas are immediate consequences of the definitions

.

A sequence { an } in a

**metric space**is a Cauchy sequence if limm . n g ( am , an )= 0 . If every Cauchy sequence is convergent , a

**metric space**is said to becomplete . The next three lemmas are immediate consequences of the definitions

.

Page 21

A space is separable , if it contains a denumerable dense set . 12 THEOREM . If a

topological space has a countable base , it is separable . Conversely , if a

A space is separable , if it contains a denumerable dense set . 12 THEOREM . If a

topological space has a countable base , it is separable . Conversely , if a

**metric****space**is separable , it has a countable base . Thus a subspace of a separable ...### What people are saying - Write a review

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero