## 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 number system ... values in the range of p , 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 number system ... values in the range of p , it will be stipulated that an

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

Page 119

Next suppose that we consider a function † ( vector or

which is defined only on the complement of a p - null set NCS . Then we say that f

is u - measurable if the function g defined by g ( s ) = f ( s ) if s ¢ N , g ( s ) = 0 if 8 ...

Next suppose that we consider a function † ( vector or

**extended**real - valued )which is defined only on the complement of a p - null set NCS . Then we say that f

is u - measurable if the function g defined by g ( s ) = f ( s ) if s ¢ N , g ( s ) = 0 if 8 ...

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 ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero