## Linear Operators, Part 1 |

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Page 143

Then the function u with domain * is known as the

o - field { * is known as the

and the measure space ( S , 2 * , u ) is the

Then the function u with domain * is known as the

**Lebesgue**extension of u . Theo - field { * is known as the

**Lebesgue**extension ( relative to u ) of the o - field E ,and the measure space ( S , 2 * , 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 lim - H ( q ) - 1

( Plu ( dq ) = 0 M ( C ) 0 ( 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 lim - H ( q ) - 1

( Plu ( dq ) = 0 M ( C ) 0 ( 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 = 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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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