## Linear Operators: General theory |

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Page 95

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

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

f a

defined ...

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

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

**valued**function andf 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

countably additive set functions ( briefly ,

able ...

**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**countably additive set functions ( briefly ,

**vector valued**measures ) than we wereable ...

Page 319

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

1 , which contradicts the assumption that ymu ... Q . E . D . In contrast to the case

of complex valued measures , the total variation of a

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

1 , which contradicts the assumption that ymu ... Q . E . D . In contrast to the case

of complex valued measures , the total variation of a

**vector valued**measure ( cf .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero