## 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 y e Y determines the linear functional f , on X defined by 1 , ( x ) = x ( y ) ...Page 477

strong , and the weak operator

strong , and the weak operator

**topologies**in B ( X , Y ) . It is evident that the uniform operator**topology**is stronger than the strong operator**topology**, and that the strong operator**topology**is stronger than the weak operator ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed 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 function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero