## Linear Operators: General theory |

### From inside the book

Results 1-3 of 45

Page 22

This contradiction proves that A is

that A is

separable . Let po be an arbitrary point of A , and let do = sup e ( po , p ) . The

number do is ...

This contradiction proves that A is

**sequentially compact**. Conversely , supposethat A is

**sequentially compact**and closed . It will first be shown that A isseparable . Let po be an arbitrary point of A , and let do = sup e ( po , p ) . The

number do is ...

Page 314

A subset K of ba ( S , E ) is weakly

a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 M ( E ) 0 uniformly for a €

K . Proof . Let K C ba ( S , E ) be weakly

A subset K of ba ( S , E ) is weakly

**sequentially compact**if and only if there existsa non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 M ( E ) 0 uniformly for a €

K . Proof . Let K C ba ( S , E ) be weakly

**sequentially compact**and let V = UT be ...Page 434

We thus conclude that K is weakly

convex set and therefore ( 3 . 13 ) X * - closed , the preceding theorem applies to

show that K is X * - compact . Q . E . D . 3 THEOREM . The weak topology of a ...

We thus conclude that K is weakly

**sequentially compact**, but since K is a closedconvex set and therefore ( 3 . 13 ) X * - closed , the preceding theorem applies to

show that K is X * - compact . Q . E . D . 3 THEOREM . The weak topology of a ...

### What people are saying - Write a review

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

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved 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