## Linear Operators: General theory |

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Page 186

The product of finite

Proof . This fact follows immediately from the formula for M ( E ) given in Corollary

4. Q.E.D. It is now easy to extend the definition of the product measure to the ...

The product of finite

**positive measure**spaces is a finite**positive measure**space .Proof . This fact follows immediately from the formula for M ( E ) given in Corollary

4. Q.E.D. It is now easy to extend the definition of the product measure to the ...

Page 212

Let u be a finite

compact metric space S. A set ACS is said to be covered in the sense of Vitali by

a family F of closed sets if each FeF has positive u - measure and there is a

positive ...

Let u be a finite

**positive measure**defined on the o - field of Borel sets of acompact metric space S. A set ACS is said to be covered in the sense of Vitali by

a family F of closed sets if each FeF has positive u - measure and there is a

positive ...

Page 725

1 n - 1 lim sup I u ( p = re ) S Ku ( e ) n + 00 nj = 0 for each set e of finite u -

measure . ( Hint . Consider the map f ( s ) → XA ( s ) / ( ps ) for each A € E with u (

A ) < 00 . ) 37 Let ( S , E , u ) be a

...

1 n - 1 lim sup I u ( p = re ) S Ku ( e ) n + 00 nj = 0 for each set e of finite u -

measure . ( Hint . Consider the map f ( s ) → XA ( s ) / ( ps ) for each A € E with u (

A ) < 00 . ) 37 Let ( S , E , u ) be a

**positive measure**space , and T a non - negative...

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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