## Linear Operators: General theory |

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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 ) = { 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

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

the single elementa then the

one of the form Paca Ec where E , is in and , for all but a finite number of a , Ec ...

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

the single elementa then the

**field**E , is the o -**field**Eq . An elementary set in S isone of the form Paca Ec where E , is in and , for all but a finite number of a , Ec ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero