## Linear Operators, Part 1 |

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Page 10

A

then a point p is a limit point , or a point of accumulation , of A provided every

set in ...

A

**neighborhood**of the set A is an open set containing A . If A is a subset of X ,then a point p is a limit point , or a point of accumulation , of A provided every

**neighborhood**of p contains at least one point q # p , with q € A . The interior of aset in ...

Page 51

If a homomorphism of one topological group into another is continuous anywhere

, it is continuous . PROOF . Let the homomorphism f : G + H be continuous at x ,

and let y eG . If V is a

...

If a homomorphism of one topological group into another is continuous anywhere

, it is continuous . PROOF . Let the homomorphism f : G + H be continuous at x ,

and let y eG . If V is a

**neighborhood**of f ( y ) , then , by Lemma 2 ( c ) , Vf ( y - 1x )...

Page 56

of the image of any

b , there is a

and so X e ...

of the image of any

**neighborhood**G of the element 0 in X contains a**neighborhood**of the element 0 in Y . Since a - 6 is a continuous function of a andb , there is a

**neighborhood**M of 0 such that M - MCG . For every x e X , xịn → 0 ,and so X e ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

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### Common terms and phrases

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