## Linear Operators: General theory |

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Page 419

Then the l '

Then the l '

**topology**of X is the**topology**obtained by taking as base all sets of the form N ( p ; A , ε ) = { 9 ||| ( p ) - | ( 9 ) / < , 1 € A } , where pe X , A is a finite subset of I , and > 0 . The terms l - open and I ' - closed ...Page 420

E in the X *

E in the X *

**topology**if and only if lim , xe = x weakly in the sense of Definition II.3.25 . On the other hand , if X is a subspace of y * , then each element ye y determines the linear functional f , on X defined by 1 , ( x ) = x ( y ) ...Page 512

is compact in the strong operator

is compact in the strong operator

**topology**. If y is also separable , A is sequentially compact in the weak operator**topology**if and only if A is compact in the weak operator**topology**. 6 If y is reflexive , then the closed unit sphere ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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