## 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 ) > \ 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 to ( s ) converges ( to fri ) ( ) ) for each s in [ a , b ] and each j = 0 , 1 , ... , p . ( c ) CP is not weakly complete , and not reflexive . ( d ) A subset A CCP is conditionally compact if and only ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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