## Linear Operators: General theory |

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Page 20

Thus A + X , and Al is open , and

1 / 2 . By assumption , the set A , does not

hence the non - void open set An S ( P1 , & 1 / 2 )

Thus A + X , and Al is open , and

**contains**a sphere S , = S ( P1 , & z ) with 0 < & <1 / 2 . By assumption , the set A , does not

**contain**the open set S ( P1 , E1 / 2 ) ;hence the non - void open set An S ( P1 , & 1 / 2 )

**contains**a sphere S , = S ( P2 ...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 – MCG . For every x € X , x / n → 0 ,

and so ...

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 – MCG . For every x € X , x / n → 0 ,

and so ...

Page 309

The set A = S - U E ; then

will now be shown that every measurable subset B of A

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

...

The set A = S - U E ; then

**contains**no atoms having measure greater than ε . Itwill now be shown that every measurable subset B of A

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

**contains**no set of with...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### 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