## Linear Operators: General theory |

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Page 95

Thus, even if the integration process is defined with respect to a scalar valued set

function p, the resulting integral v may be a

desirable therefore to formulate some of the basic and elementary concepts in

such a ...

Thus, even if the integration process is defined with respect to a scalar valued set

function p, the resulting integral v may be a

**vector valued**set function. It isdesirable therefore to formulate some of the basic and elementary concepts in

such a ...

Page 318

last section permit us to develop a more satisfactory theory of

- ably additive set functions (briefly,

**Vector Valued**Measures The theorems on spaces of set functions proved in thelast section permit us to develop a more satisfactory theory of

**vector valued**count- ably additive set functions (briefly,

**vector valued**measures) than we were able ...Page 319

A vector measure is bounded, and the set {x*n\x* e X*, \x*\ 1} of numerical

measures is weakly sequentially compact as a subset of ... Q.E.D. In contrast to

the case of complex valued measures, the total variation of a

measure (cf.

A vector measure is bounded, and the set {x*n\x* e X*, \x*\ 1} of numerical

measures is weakly sequentially compact as a subset of ... Q.E.D. In contrast to

the case of complex valued measures, the total variation of a

**vector valued**measure (cf.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

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### 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