## 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 an. 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 ,Ťa(-Ea) it is convenient to assume that for all but a

Ea) ...

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

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

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

Page 579

Every non-zero

such a

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

, the ...

Every non-zero

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

**number**X, the projection E(X) has 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 84 | 34 |

Copyright | |

80 other sections not shown

### 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-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