## Linear Operators, Part 1 |

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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

. If every Cauchy sequence is convergent , a

The next three lemmas are immediate consequences of the definitions .

A sequence { an } in a

**metric space**is a Cauchy sequence if limm , glam , an ) = 0. If every Cauchy sequence is convergent , a

**metric space**is said to be complete .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 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero