## Linear Operators: General theory |

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Page 54

A linear mapping of one F - space into another is continuous if and only if it maps

A linear mapping of one F - space into another is continuous if and only if it maps

**bounded**sets into**bounded**sets . PROOF . Let X , Y be F - spaces , let T : X → Y be linear and continuous , and let B C X be**bounded**.Page 97

and to be

and to be

**bounded**if u is**bounded**. The total variation v ( u ) of an additive set function u is important because it dominates u in the sense that v ( u , E ) 2 ( u ( E ) for E € £ ; the reader should test his comprehension of ...Page 345

if and only if it is

if and only if it is

**bounded**and thi ( s ) converges ( to fli ) ( s ) ) for each 8 in [ a , b ] and each i = 0 , 1 , ... , p . ( c ) ( p is not weakly complete , and not reflexive . ( d ) A subset A Ç CP is conditionally compact if and ...### 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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### 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