## Linear Operators: General theory |

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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 ...

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 ...

Page 294

Since K is weakly

weakly , and it may be assumed without loss of generality that { / n } itself

converges weakly . By Theorem 7 , { Sein ( s ) u ( ds ) } converges for each Ee E .

Let 1 ...

Since K is weakly

**sequentially compact**, a subsequence of { { n } convergesweakly , and it may be assumed without loss of generality that { / n } itself

converges weakly . By Theorem 7 , { Sein ( s ) u ( ds ) } converges for each Ee E .

Let 1 ...

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 H ( E ) 0 unifarmly 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 H ( E ) 0 unifarmly for a €

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

**sequentially compact**and let V = UT be ...### What people are saying - Write a review

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

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