## Linear Operators, Part 1 |

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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 233

All of these integrals are

present a Lebesgue type theory of integration of a scalar function with respect to

a

...

All of these integrals are

**countably additive**integrals . In Section IV . 10 we willpresent a Lebesgue type theory of integration of a scalar function with respect to

a

**countably additive**vector valued measure . Instances in which both the function...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 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 obtained 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