## 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 to ...

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 to ...

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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero