## Linear Operators, Part 1 |

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Page 136

There is a uniquely determined smallest

o -

namely the

intersection ...

There is a uniquely determined smallest

**field**and a uniquely determined smallesto -

**field**containing a given family of sets . Proof . There is at least one**field**,namely the

**field**of all subsets of S , which contains a given family t . Theintersection ...

Page 166

If we put E ( E ) = { F€ E 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 ) = { F€ E 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 Στο E . If Σ , is a**field**, Εε Σ . , and ...Page 201

The symbol will be used for the o -

of all sets in S of the form S . XE with Ε ε Σ will be denoted by Σπ . 18 LEMMA .

For each r the family Sx is a

identities ...

The symbol will be used for the o -

**field**of sets in S determined by £ . The familyof all sets in S of the form S . XE with Ε ε Σ will be denoted by Σπ . 18 LEMMA .

For each r the family Sx is a

**field**of sets in S and £ = U , Sn . PROOF . Theidentities ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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