## Linear Operators: General theory |

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Page 10

A

a point p is a Zimi< point, or a point o/ accumulation, of ^4 provided every

set in ...

A

**neighborhood**of the set A is an open set containing A. If A is a subset of X, thena point p is a Zimi< point, or a point o/ accumulation, of ^4 provided every

**neighborhood**of p contains at least one point q ^ p, with o; c .4. The interior of aset in ...

Page 51

Let the homomorphism / :G ->fl be continuous at x, and let y e G. If V is a

is a

such that ...

Let the homomorphism / :G ->fl be continuous at x, and let y e G. If V is a

**neighborhood**of f(y), then, by Lemma 2(c), Vf(y~1x) is a**neighborhood**of f(x). If Vis a

**neighborhood**of x such that f(V)Q Vfiy^x), then Ux^y is a**neighborhood**of ysuch that ...

Page 56

of the image of any

and b, there is a

>• 0, and so ...

of the image of any

**neighborhood**G of the element 0 in X contains a**neighborhood**of the element 0 in 9). Since a— b is a continuous function of aand b, there is a

**neighborhood**M of 0 such that M — M Q G. For every x e S, x/n ->• 0, and so ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero