## Linear Operators: General theory |

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Page 164

Since the extension of a

follows that {/<„(£)} is abounded non-decreasing set of real numbers for each E e

Zv We define = lim„ jHn(E), E e Zv By Corollary 4, Xx is countably additive on Zv ...

Since the extension of a

**non**-**negative**set function on Z to Zx is**non**-**negative**, itfollows that {/<„(£)} is abounded non-decreasing set of real numbers for each E e

Zv We define = lim„ jHn(E), E e Zv By Corollary 4, Xx is countably additive on Zv ...

Page 179

4 Theorem. Let (S, E, fi) be a positive measure space, f a

measurable function defined on S and l(E)=jEf(s)v(ds), EeZ. Let g be a

s)k(ds) = jE ...

4 Theorem. Let (S, E, fi) be a positive measure space, f a

**non**-**negative**^.-measurable function defined on S and l(E)=jEf(s)v(ds), EeZ. Let g be a

**non**-**negative**X-measurable function defined on S. Then fg is u- measurable, and j£ g(s)k(ds) = jE ...

Page 314

equation X1{F) = p^EF), F e Ev Then U(X1)(G) = (UUl)(GE) for G in E2. By part (b)

, v(U(^), E) = |£7(^)| = %\ = v(uv E). Q.E.D. 12 Theorem. A subset K of ba(S, E) is

weakly sequentially compact if arid only if there exists a

equation X1{F) = p^EF), F e Ev Then U(X1)(G) = (UUl)(GE) for G in E2. By part (b)

, v(U(^), E) = |£7(^)| = %\ = v(uv E). Q.E.D. 12 Theorem. A subset K of ba(S, E) is

weakly sequentially compact if arid only if there exists a

**non**-**negative**u in ba(S, ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero