## Linear Operators: General theory |

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Page 126

results of the preceding sections can be considerably extended . DEFINITION .

**Countably Additive**Set Functions The basis for the present section is a**countably****additive**set function defined on a o - field of subsets of a set . In this case theresults of the preceding sections can be considerably extended . DEFINITION .

Page 132

This lemma shows that if each member of a generalized sequence { 2x } of finite ,

where a is also a finite ,

continuous ...

This lemma shows that if each member of a generalized sequence { 2x } of finite ,

**countably additive**measures is u - continuous and if lim , 12 ( 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 has a

- field determined by E. 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 has a

**countably additive**non - negative extension to the o- field determined by E. If u is o - finite on then this extension is unique . Proof .

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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