## Linear Operators: General theory |

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Page 20

Thus A , X , and A is open , and

2 . By assumption , the set A , does not

hence the non - void open set An S ( P1 , & z / 2 )

Ɛg ) ...

Thus A , X , and A is open , and

**contains**a sphere S , = S ( P . , E ) with 0 < & < 1/2 . By assumption , the set A , does not

**contain**the open set S ( P1 , € 1 / 2 ) ;hence the non - void open set An S ( P1 , & z / 2 )

**contains**a sphere S , = S ( P2 ,Ɛg ) ...

Page 56

of the image of any neighborhood G of the element 0 in X

neighborhood of the element 0 in Y. Since a - b is a continuous function of a and

b , there is a neighborhood M of 0 such that M - MĒG . For every x e X , x / n → 0 ,

and so x e ...

of the image of any neighborhood G of the element 0 in X

**contains**aneighborhood of the element 0 in Y. Since a - b is a continuous function of a and

b , there is a neighborhood M of 0 such that M - MĒG . For every x e X , x / n → 0 ,

and so x e ...

Page 309

It will now be shown that every measurable subset B of A

u ( F ) S £ . Suppose , on the contrary , that some such set B

with positive measure at most ε . Then B is not an atom and thus

...

It will now be shown that every measurable subset B of A

**contains**a set F with 0 <u ( F ) S £ . Suppose , on the contrary , that some such set B

**contains**no set ofwith positive measure at most ε . Then B is not an atom and thus

**contains**a set G...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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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