## Linear Operators: General theory |

### From inside the book

Results 1-3 of 80

Page 419

Nelson Dunford, Jacob T. Schwartz, William G. Bade. 2 DEFINITION . Let X be a

linear vector space , and let I be a total subspace of X * . Then the I

is the

Nelson Dunford, Jacob T. Schwartz, William G. Bade. 2 DEFINITION . Let X be a

linear vector space , and let I be a total subspace of X * . Then the I

**topology**of Xis the

**topology**obtained by taking as base all sets of the form N ( p ; A , ε ) = { 9 ...Page 420

Nelson Dunford, Jacob T. Schwartz, William G. Bade. in the X *

only if lim , Xx = x weakly in the sense of Definition II . 3 . 25 . On the other hand ,

if X is a subspace of yî , then each element y e Y determines the linear functional f

...

Nelson Dunford, Jacob T. Schwartz, William G. Bade. in the X *

**topology**if andonly if lim , Xx = x weakly in the sense of Definition II . 3 . 25 . On the other hand ,

if X is a subspace of yî , then each element y e Y determines the linear functional f

...

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

User Review - Flag as inappropriate

i want to read

### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

10 other sections not shown

### Other editions - View all

### Common terms and phrases

Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero