Linear Operators, Part 1 |
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Page 136
( Hahn extension ) Every countably additive non - negative extended real valued
set function u on a field E 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 .
( Hahn extension ) Every countably additive non - negative extended real valued
set function u on a field E 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 .
Page 143
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 ...
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
Extension of Linear Transformation . Taylor [ 1 ] studied conditions under which
the extension of linear functionals will be a uniquely defined operation . Kakutani
[ 6 ] showed that this operation is linear and isometric for each closed linear ...
Extension of Linear Transformation . Taylor [ 1 ] studied conditions under which
the 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 | |
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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