## Linear Operators: General theory |

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Page 136

There is a uniquely

o - 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 . 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 . The

intersection ...

Page 455

Let G = U_ Yn ; then f is

subset of X * , there exists a denumerable subset G , of X * such that each feF is

can be ...

Let G = U_ Yn ; then f is

**determined**by G. It follows that if F is a denumerablesubset of X * , there exists a denumerable subset G , of X * such that each feF is

**determined**by GF We can even assert that each continuous linear functional fcan be ...

Page 657

... determinism whereby it is known that an initial state x will , after t seconds have

elapsed , have passed into a uniquely

uniquely

) .

... determinism whereby it is known that an initial state x will , after t seconds have

elapsed , have passed into a uniquely

**determined**new state y . Since y isuniquely

**determined**by x and t , a function , on S to S is defined by the equa$ :( «) .

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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