## Linear Operators: General theory |

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Page 164

Since the extension of a

follows that { ūn ( E ) } is a bounded non - decreasing set of real numbers for each

Ee & . We define 2 ( E ) = limnīn ( E ) , E € En . By Corollary 4 , 2 , is countably ...

Since the extension of a

**non**-**negative**set function on Eto E , is**non**-**negative**, itfollows that { ūn ( E ) } is a bounded non - decreasing set of real numbers for each

Ee & . We define 2 ( E ) = limnīn ( E ) , E € En . By Corollary 4 , 2 , is countably ...

Page 179

Let ( S , E , u ) be a positive measure space , f a

function defined on S and 2 ( E ) = f ( s ) u ( ds ) , EcE . Let g be a

- measurable function defined on S . Then fg is u - measurable , and s ) g ( s ) u (

ds ) ...

Let ( S , E , u ) be a positive measure space , f a

**nonnegative**u - measurablefunction defined on S and 2 ( E ) = f ( s ) u ( ds ) , EcE . Let g be a

**non**-**negative**2- measurable function defined on S . Then fg is u - measurable , and s ) g ( s ) u (

ds ) ...

Page 314

A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists

a

K . Proof . Let K C ba ( S , E ) be weakly sequentially compact and let V = UT be ...

A subset K of ba ( S , E ) is weakly sequentially compact if and only if there exists

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 sequentially compact and let V = UT be ...

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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