## Linear Operators: General theory |

### From inside the book

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Page 137

11 DEFINITION . An additive set function u defined on a

topological space S is said to be regular if for each Ee and a > 0 there is a set F in

whose closure is contained in E and a set

that lu ...

11 DEFINITION . An additive set function u defined on a

**field**of subsets of atopological space S is said to be regular if for each Ee and a > 0 there is a set F in

whose closure is contained in E and a set

**G**in whose interior contains E suchthat lu ...

Page 166

If we put E ( E ) = { FeE | F C E } it is clear that E ( E ) is a

that E ( E ) is the family of all sets AE , A ...

f ( s ) =

If we put E ( E ) = { FeE | F C E } it is clear that E ( E ) is a

**field**of subsets of E , andthat E ( E ) is the family of all sets AE , A ...

**g**both contain the set E , the statementf ( s ) =

**g**( s ) u - almost everywhere on E means that there is a set A Ç E with v ...Page 292

Let & be a o -

a sequence of countably additive set ... then F , F , ε Σχ . It is also clear that if F , ε

Σχ , then S - F4 € £3 , and that if F1 , F2€ £ , with F _ F , = $ , then Fi UF , c £

Let & be a o -

**field**of sets , E , a subfield of Edetermining the o -**field**E , and { un }a sequence of countably additive set ... then F , F , ε Σχ . It is also clear that if F , ε

Σχ , then S - F4 € £3 , and that if F1 , F2€ £ , with F _ F , = $ , then Fi UF , c £

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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