## Linear Operators, Part 1 |

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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

E € £ . We define ( E ) = limnûn ( E ) , E €7 . By Corollary 4 , , is countably ...

Since the extension of a

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

E € £ . We define ( E ) = limnûn ( E ) , E €7 . By Corollary 4 , , is countably ...

Page 179

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

measurable function defined on S and 1 ( E ) = Set ( s ) u ( ds ) , Ee2 . Let g be a

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

**nonnegative**u -measurable function defined on S and 1 ( E ) = Set ( s ) u ( ds ) , Ee2 . Let g be a

**non**-**negative**2 - measurable function defined on S . Then fg is u - measurable ...Page 314

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

a

Κ . Proof . Let KC 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 ul E ) →0 uniformly for λεΚ . Proof . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

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