## 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 ) = { F € | F CE } it is clear that E ( E ) is a

that E ( E ) is the family of all sets AE , A € E , and that if Eis a o -

is a o -

If we put E ( E ) = { F € | F CE } 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 Eis 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 Σ ...Page 201

The symbol will be used for the o -

identities ( S , XE , U ( S , AF , ) = S , X ( ENU F , ) , and ( S . xxx E ) = SqXEA show

that { is a

The symbol will be used for the o -

**field**of sets in S determined by £ . ... Theidentities ( S , XE , U ( S , AF , ) = S , X ( ENU F , ) , and ( S . xxx E ) = SqXEA show

that { is a

**field**. It is clear from its definition that £ , contains all of the**fields**{ n .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero