## Linear Operators: General theory |

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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 , E * , 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 , E * , 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 1 lim \ / ( q ) -- | ( p ) \ n ( dq )

= 0 M ( C ) → 0 M ( C ) JC 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 1 lim \ / ( q ) -- | ( p ) \ n ( dq )

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

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

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

Sög ( s ) dh ( s ) exists . Let f be a continuous increasing function on an open

interval ( c , d ) , with f ( c ) = a , f ( d ) = b . Show that the

integral ...

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

**Lebesgue**- Stieltjes integral ISög ( s ) dh ( s ) exists . Let f be a continuous increasing function on an open

interval ( c , d ) , with f ( c ) = a , f ( d ) = b . Show that the

**Lebesgue**- Stieltjesintegral ...

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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 Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero