## Linear Operators: General theory |

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Page 22

in A has a neighborhood containing at most a

in A has a neighborhood containing at most a

**finite number**of a„. Since a**finite****number**of these neighborhoods cover A, the sequence {an} consists of only a**finite number**of distinct points of A, and therefore most certainly does have a ...Page 200

In order to avoid difficulties which might otherwise arise from the presence of

infinite products such as JJ jua(Ea) it is convenient to assume that for all but a

Ea) is ...

In order to avoid difficulties which might otherwise arise from the presence of

infinite products such as JJ jua(Ea) it is convenient to assume that for all but a

**finite number**of a, //a is non-negative and fia(Sa) = 1. In this case the product ux(Ea) is ...

Page 579

Every nan-zero

such a

given by the formula where v is the order of the pole. Proof. Since a(T) is compact

, the ...

Every nan-zero

**number**in a(T) is a pole of T and has**finite**positive index. Forsuch a

**number**X, the projection E(X) lias a non-zero**finite**dimensional rangegiven by the formula where v is the order of the pole. Proof. Since a(T) is compact

, the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero