## Linear Operators: General theory |

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Page 132

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

where 2 is also a finite ,

continuous .

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

**countably additive**measures is u - continuous and if lima da ( E ) = 2 ( E ) , Ec£ ,where 2 is also a finite ,

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

Page 136

( Hahn extension ) Every

set function u on a field E has a

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

( 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 E . If u is o - finite on then this extension is unique .

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 | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero