## Linear Operators: General theory |

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Page 136

6 LEMMA . There is a uniquely determined smallest

determined smallest o -

at least one

family t .

6 LEMMA . There is a uniquely determined smallest

**field**and a uniquelydetermined smallest o -

**field**containing a given family of sets . PROOF . There isat least one

**field**, namely the**field**of all subsets of S , which contains a givenfamily t .

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 € E , and that if & is a o -

) is a o -

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 € E , and that if & is a o -

**field**, then E ( E) is a o -

**field**. Σ ( Ε ) is called the restriction of Σ to E . If Σ , is a**field**, Εε Σ . , and ...Page 201

The family of elementary sets in S will be denoted by E and the

determined by E will be denoted by £ . The symbol will be used for the o -

sets in S determined by E . The family of all sets in S of the form Sq , XER with Ε ε

...

The family of elementary sets in S will be denoted by E and the

**field**of sets in Sdetermined by E will be denoted by £ . The symbol will be used for the o -

**field**ofsets in S determined by E . The family of all sets in S of the form Sq , XER with Ε ε

...

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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