## Linear Operators: General theory |

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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**. If X is a topological space such that there exists a metric function whose topology is the same ...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 complete . The next three lemmas are immediate consequences of the definitions .Page 21

A subset A of a topological space X is sequentially compact , if every sequence of points in A has a subsequence converging to a point of X. 11. DEFINITION . ... Conversely , if a

A subset A of a topological space X is sequentially compact , if every sequence of points in A has a subsequence converging to a point of X. 11. DEFINITION . ... Conversely , if a

**metric space**is separable , it has a countable base .### 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 | |

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