## Linear Operators: General theory |

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Page 10

A

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

set in X ...

A

**neighborhood**of the set A is an open set containing A. If A is a subset of X, thena 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 e A. The interior of aset in X ...

Page 51

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

is a

y ...

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

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

**neighborhood**of x such that f(V)Q Vf(y~1x), then Ux~h) is a**neighborhood**ofy ...

Page 56

of the image of any

and b, there is a

0, 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 '?). Since a — b is a continuous func tion of aand b, there is a

**neighborhood**M of 0 such that M~J\I Q G For every x (S, xjn -*□0, and so x e ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact