## 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

linear and continuous , and let B C X be

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 belinear and continuous , and let B C X be

**bounded**. For every neighborhood V of ...Page 345

if and only if it is

and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

a ...

if and only if it is

**bounded**and ( s ) converges ( to flo ( s ) ) for each s in [ a , b ]and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

**bounded**and for eacha ...

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9 ( 11 )

operator , definition , II . 3 . 5 ( 60 ) set in a linear topological space , II . 1 . 7 ( 51 )

criterion for boundedness in a B - space , II . 3 . 3 ( 59 ) remarks on , ( 80 ) totally ...

9 ( 11 )

**Bounded**, essentially ( or u - essentially ) , definition , III . 1 . 11 ( 100 )operator , definition , II . 3 . 5 ( 60 ) set in a linear topological space , II . 1 . 7 ( 51 )

criterion for boundedness in a B - space , II . 3 . 3 ( 59 ) remarks on , ( 80 ) totally ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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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 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 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