## 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 | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero