## Linear Operators: General theory |

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Page 143

... with domain 2 * is known as the

known as the

measure space ( S , 3 * , u ) is the

E , u ) .

... with domain 2 * is known as the

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

**Lebesgue**extension ( relative to u ) of the o - field E , and themeasure space ( S , 3 * , u ) is the

**Lebesgue**extension of the measure space ( S ,E , u ) .

Page 218

Let | be a vector valued

Euclidean n - space . The set of all points p at which lim cil 11 ( 9 ) - 1 ( p ) \ u ( dq

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

Let | be a vector valued

**Lebesgue**integrable function defined on an open set inEuclidean n - space . The set of all points p at which lim cil 11 ( 9 ) - 1 ( p ) \ u ( dq

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

**Lebesgue**set of the function f . Clearly ...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 = Sag ( 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 = Sag ( s ) dh ( s ) exists . Let f be a continuous increasing

function ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero