## Linear Operators: General theory |

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Page 95

In some cases that will be encountered the values of /x are not scalars, but

customarily where integration is used in this text fi is a scalar

/ a

defined ...

In some cases that will be encountered the values of /x are not scalars, but

customarily where integration is used in this text fi is a scalar

**valued**function and/ a

**vector**(or scalar)**valued**function. Thus, even if the integration process isdefined ...

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

4, the set {y*u} of scalar valued measures is uniformly countably additive on Ev

which contradicts the assumption that y^ji(En ) > e, ... Q.E.D. In contrast to the

case of complex valued measures, the total variation of a

(cf.

4, the set {y*u} of scalar valued measures is uniformly countably additive on Ev

which contradicts the assumption that y^ji(En ) > e, ... 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 | |

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions 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 Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact