## 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 345

if and only if it is

if and only if it is

**bounded**and thy ( s ) converges ( to f ( * ) ( s ) ) for each 8 in [ a , b ] and each j = 0 , 1 , ... , p .Page 838

9 ( 11 )

9 ( 11 )

**Bounded**, essentially ( or u - essentially ) , definition , III . 1 . 11 ( 100 ) operator , definition , II . 3 .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero