## Linear Operators, Part 1 |

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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

Suppose that E is a set in E. If we put E ( E ) = { F € E | F CE } it is clear that E ( E )

is a

that if is a o -

If ...

Suppose that E is a set in E. If we put E ( E ) = { F € E | F CE } it is clear that E ( E )

is a

**field**of subsets of E , and that E ( E ) is the family of all sets AE , A € E , andthat if is a o -

**field**, then ( E ) is a o -**field**. Σ ( Ε ) is called the restriction of Σ to E.If ...

Page 201

There should be no confusion between the symbols Ez and Eq , for if a reduces

to the single elementa then the

is one of the form P & EA E where Eq is in E , and , for all but a finite number of o

...

There should be no confusion between the symbols Ez and Eq , for if a reduces

to the single elementa then the

**field**E , is the o -**field**Eq . An elementary set in Sis one of the form P & EA E where Eq is in E , and , for all but a finite number of o

...

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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