## Linear Operators: General theory |

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Results 1-3 of 73

Page 186

The product of finite

Proof. This fact follows immediately from the formula for /i(E) given in Corollary 4.

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

The product of finite

**positive measure spaces**is a finite**positive measure space**.Proof. This fact follows immediately from the formula for /i(E) given in Corollary 4.

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

Page 302

Let (S, 27, fi) be a

LP(S, 27, //), 1 ^ p < oo, is a complete lattice. Proof. It is evidently sufficient to

show that if {/„} is a set of functions in Lj such that 0 fa g0 for some g0 e Lv then

supa ...

Let (S, 27, fi) be a

**positive measure space**. Then the real partially ordered spaceLP(S, 27, //), 1 ^ p < oo, is a complete lattice. Proof. It is evidently sufficient to

show that if {/„} is a set of functions in Lj such that 0 fa g0 for some g0 e Lv then

supa ...

Page 725

I n-1 lim sup - ^ fl{(F~,e) KMe) n-*oo n j-0 for each set e of finite /^-measure. (Hint.

Consider the map f(s) □*□ XA(s)f(fs) f°T eacn ^ € ^ w'tn M-d) < °°0 37 Let (5, 2T,

// ) be a

I n-1 lim sup - ^ fl{(F~,e) KMe) n-*oo n j-0 for each set e of finite /^-measure. (Hint.

Consider the map f(s) □*□ XA(s)f(fs) f°T eacn ^ € ^ w'tn M-d) < °°0 37 Let (5, 2T,

// ) be a

**positive measure space**, and T a non-negative linear transformation of ...### 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 |

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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