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

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

) is a o -

is ...

If we put E ( E ) = { F € E | F ÇE } 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 , and that if is a o -

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

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

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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

Acad 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 semi-group 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