## Linear Operators, Part 1 |

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Page 95

Finitely Additive Set

process is

integral v ...

Finitely Additive Set

**Functions**In contrast to the terms real**function**, complex**function**, etc. , where the adjectives real and ... Thus , even if the integrationprocess is

**defined**with respect to a scalar valued set**function**u , the resultingintegral v ...

Page 98

The total variation of an additive set

set S is also additive on E. Proof . Let { A } } be a finite sequence of disjoint sets in

E with A , CEUF where E , F € £ and EF = $ . Let E ; EA , Fi = FA , then Elu ( A ) ...

The total variation of an additive set

**function defined**on a field E of subsets of aset S is also additive on E. Proof . Let { A } } be a finite sequence of disjoint sets in

E with A , CEUF where E , F € £ and EF = $ . Let E ; EA , Fi = FA , then Elu ( A ) ...

Page 182

Q.E.D. Note : The unique u - integrable

the Radon - Nikodım derivative of 2 with respect to u and is often denoted by da /

du . Thus dh / du is

...

Q.E.D. Note : The unique u - integrable

**function**f of Theorems 2 and 7 is calledthe Radon - Nikodım derivative of 2 with respect to u and is often denoted by da /

du . Thus dh / du is

**defined**u - almost everywhere by the formula ( da 2 ( E ) = Εε...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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 condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero