## 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

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

measurable function defined on S and s ) u ( ds ) , E € £ . 20 Let g be a

Seg ( s ) ...

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

**nonnegative**u -measurable function defined on S and s ) u ( ds ) , E € £ . 20 Let g be a

**non**-**negative**a - measurable function defined on S . Then fg is l - measurable , andSeg ( s ) ...

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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

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