## Linear Operators: General theory |

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Page 95

CHAPTER III Integration and

CHAPTER III Integration and

**Set Functions**1. Finitely Additive**Set Functions**In contrast to the terms real function , complex function , etc. , where the ...Page 96

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**set function**u defined on a family t of sets is said to be additive or finitely additive if u ( t ) = 0 and u ( A U AZ .Page 837

A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely continuous functions , definition , IV.2.22 ( 242 )

A on Abel summability , of series , II.4.42 ( 76 ) Abelian group , ( 34 ) Absolutely continuous functions , definition , IV.2.22 ( 242 )

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero