## Linear Operators: General theory |

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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 ...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 ...Page 345

if and only if it is

if and only if it is

**bounded**and foll ( s ) converges ( to fly ( s ) ) for each s in [ a , b ] and each j = 0 , 1 , ... , p .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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