## Linear Operators: General theory |

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Page 95

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

real ... necessary to add values in the range of u , it will be stipulated that an

- 00 .

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

**extended**real ... necessary to add values in the range of u , it will be stipulated that an

**extended**real valued set function has at most one of the improper values on and- 00 .

Page 126

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

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

valued additive set function defined on a field of subsets of a set S . Then u is

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**realvalued additive set function defined on a field of subsets of a set S . Then u is

said to ...

Page 137

The total variation of a regular additive complex or

function on a field is regular . Moreover , the positive and negative variations of a

bounded regular real valued additive set function are also regular . Proof . If u is a

...

The total variation of a regular additive complex or

**extended**real valued setfunction on a field is regular . Moreover , the positive and negative variations of a

bounded regular real valued additive set function are also regular . Proof . If u is a

...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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