## Linear Operators: General theory |

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Page 186

Q.E.D. 3 Definition. The

called the product

proof of Theorem 2 we showed that the characteristic function of every set E e Z =

Zx X . .

Q.E.D. 3 Definition. The

**measure space**(S,Z,fi) constructed in Theorem 2 iscalled the product

**measure space**of the**measure spaces**In the course of theproof of Theorem 2 we showed that the characteristic function of every set E e Z =

Zx X . .

Page 188

Q.E.D. As in the case of finite

S, E, u) constructed in Corollary 6 from the a-finite

product

...

Q.E.D. As in the case of finite

**measure spaces**we shall call the**measure space**(S, E, u) constructed in Corollary 6 from the a-finite

**measure spaces**(S^E^fa) theproduct

**measure space**and write (S, E, ft) = The best known example of Theorem...

Page 405

sional Gauss

real sequences x = [xt] as distinct from l2, which is the subspace of s determined

by the condition Jill37/ < 00 • The

sional Gauss

**measure**on the real line. The**space**s is, of course, the**space**of allreal sequences x = [xt] as distinct from l2, which is the subspace of s determined

by the condition Jill37/ < 00 • The

**measure**/u^ is countably additive; the subset ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero