## Linear Operators: General theory |

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Page 186

Thus , the field of Lemma 1 is a o - field and the measure u of Lemma 1 is

countably additive on Ø . Consequently , the restriction of u to the o ... Let ( S , E ,

u ) be the product of finite

M2 ) .

Thus , the field of Lemma 1 is a o - field and the measure u of Lemma 1 is

countably additive on Ø . Consequently , the restriction of u to the o ... Let ( S , E ,

u ) be the product of finite

**positive measure**spaces ( S1 , E1 , M ) and ( S2 , E2 ,M2 ) .

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

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