## 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

The metric topology in X is the smallest topology containing the spheres . The set X , with its metric topology , is called a

**metric space**.Page 20

A sequence { an } in a

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 ...Page 21

Conversely , if a

Conversely , if a

**metric space**is separable , it has a countable base . Thus a subspace of a separable**metric space**is separable . Proof .### 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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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