## 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 olam , 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 | 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