## Linear Operators, Part 1 |

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Page 143

Then the function u with domain 2 * is known as the

The o - field { * is known as the

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

Then the function u with domain 2 * is known as the

**Lebesgue**extension of u .The o - field { * 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

9 DEFINITION . Let f be a vector valued

an open set in Euclidean n - space . The set of all points p at which lima l ( q ) - 1 (

p ) ( dq ) = 0 M ( C ) →0 u ( C ) JC is called the

9 DEFINITION . Let f be a vector valued

**Lebesgue**integrable function defined onan open set in Euclidean n - space . The set of all points p at which lima l ( q ) - 1 (

p ) ( dq ) = 0 M ( C ) →0 u ( C ) JC is called the

**Lebesgue**set of the function f .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 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad 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 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 transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero