## 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 , 8 ) = { 9 | \ / ( p ) - [ ( 9 ) < € , 1 € A } , where pe X , A is a finite subset of T ' , and ε > 0 .Page 420

in the X *

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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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