## Linear Operators: General theory |

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Page 132

This lemma shows that if each member of a generalized sequence { aa } of finite ,

where a is also a finite ,

continuous ...

This lemma shows that if each member of a generalized sequence { aa } of finite ,

**countably additive**measures is u - continuous and if lim , 2 . ( E ) = 2 ( E ) , Ee ,where a is also a finite ,

**countably additive**measure , then 2 is also u -continuous ...

Page 136

( Hahn extension ) Every

set function u on a field E has a

the o - field determined by £ . If u is o - finite on then this extension is unique .

Proof .

( Hahn extension ) Every

**countably additive**non - negative extended real valuedset function u on a field E has a

**countably additive**non - negative extension tothe o - field determined by £ . If u is o - finite on then this extension is unique .

Proof .

Page 318

Vector Valued Measures The theorems on spaces of set functions proved in the

last section permit us to develop a more satisfactory theory of vector valued

able ...

Vector Valued Measures The theorems on spaces of set functions proved in the

last section permit us to develop a more satisfactory theory of vector valued

**countably additive**set functions ( briefly , vector valued measures ) than we wereable ...

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### 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