## Linear Operators, Part 1 |

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Page 136

( Hahn

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

the o - 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 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 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 , E * , ) 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 , E * , ) 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

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero