## Linear Operators: General theory |

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Page 136

( Hahn

set function u on a field has a countably additive non - negative

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

( 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 .Page 143

Then the function u with domain £ * is known as the Lebesgue

The o - field * is known as the Lebesgue

, and the measure space ( S , * , u ) is the Lebesgue

Then the function u with domain £ * is known as the Lebesgue

**extension**of u .The o - field * is known as the Lebesgue

**extension**( relative to u ) of the o - field E, and the measure space ( S , * , u ) is the Lebesgue

**extension**of the measure ...Page 554

the

[ 6 ] showed that this operation is linear and isometric for each closed linear ...

**Extension**of Linear Transformation . Taylor ( 1 ) studied conditions under whichthe

**extension**of linear functionals will be a uniquely defined operation . Kakutani[ 6 ] showed that this operation is linear and isometric for each closed linear ...

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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