## Linear Operators: General theory |

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Page 143

Then the function u with domain & * is known as the

The o - field 2 * is known as the

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

Then the function u with domain & * is known as the

**Lebesgue**extension of u .The o - field 2 * is known as the

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

**Lebesgue**extension of the measure ...Page 218

Let f be a vector valued

Euclidean n - space . The set of all points p at which lim M ( C ) 0 nic , S. 195 – + (

p ) ( da ) = 0 is called the

Let f be a vector valued

**Lebesgue**integrable function defined on an open set inEuclidean n - space . The set of all points p at which lim M ( C ) 0 nic , S. 195 – + (

p ) ( da ) = 0 is called the

**Lebesgue**set of the function f . Clearly , the**Lebesgue**...Page 223

5 Let h be a function of bounded variation on the interval ( a , b ) and continuous

on the right . Let g be a function defined on ( a , b ) such that the

Stieltjes integral I = Såg ( s ) dh ( s ) exists . Let f be a continuous increasing

function ...

5 Let h be a function of bounded variation on the interval ( a , b ) and continuous

on the right . Let g be a function defined on ( a , b ) such that the

**Lebesgue**-Stieltjes integral I = Såg ( s ) dh ( s ) exists . Let f be a continuous increasing

function ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero