## Linear Operators: General theory |

### From inside the book

Results 1-3 of 76

Page 419

Then the l '

form N ( p ; A , ε ) = { q | | | ( p ) - | ( 9 ) | < E , JE A } , where pe X , A is a finite

subset of I , and a > 0 . The terms I - open and I - closed subsets of X , -

continuous ...

Then the l '

**topology**of X is the**topology**obtained by taking as base all sets of theform N ( p ; A , ε ) = { q | | | ( p ) - | ( 9 ) | < E , JE A } , where pe X , A is a finite

subset of I , and a > 0 . The terms I - open and I - closed subsets of X , -

continuous ...

Page 420

Nelson Dunford, Jacob T. Schwartz. in the 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

...

Nelson Dunford, Jacob T. Schwartz. in the X *

**topology**if and only if lim , xe = xweakly 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

...

Page 512

is compact in the strong operator

sequentially compact in the weak operator

the weak operator

X , Y ) is ...

is compact in the strong operator

**topology**. If Y is also separable , A issequentially compact in the weak operator

**topology**if and only if A is compact inthe weak operator

**topology**. 6 If Y is reflexive , then the closed unit sphere of B (X , Y ) is ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero