## Linear Operators: General theory |

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Page 95

... as scalar

deeper properties of vector

is desirable also to allow the set function u to have its values in the

...

... as scalar

**valued**set functions may be studied . A discussion of some of thedeeper properties of vector

**valued**set functions is contained in Section IV . 10 . Itis desirable also to allow the set function u to have its values in the

**extended real**...

Page 126

In this case the results of the preceding sections can be considerably extended .

1 DEFINITION . Let u be a vector valued , complex valued , or

said ...

In this case the results of the preceding sections can be considerably extended .

1 DEFINITION . Let u be a vector valued , complex valued , or

**extended real****valued**additive set function defined on a field & of subsets of a set S . Then u issaid ...

Page 137

Q . E . D . The next results on the extension of measures make use of interesting

relations between the topology of a space and certain ... The total variation of a

regular additive complex or

.

Q . E . D . The next results on the extension of measures make use of interesting

relations between the topology of a space and certain ... The total variation of a

regular additive complex or

**extended real valued**set function on a field is regular.

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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