## Linear Operators: General theory |

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Page 224

It will be assumed that the reader is familiar with the elementary theory of

complex valued

applied here to obtain the extensions which will be used later. Throughout this

section ...

It will be assumed that the reader is familiar with the elementary theory of

complex valued

**analytic**functions of one complex variable and this theory will beapplied here to obtain the extensions which will be used later. Throughout this

section ...

Page 229

A function /

expansion /(«) = f «,(«-*)», which converges uniformly and absolutely in every

annulus a+e :£ |z— z0| £S /S — e with e > 0. The coefficients ap are given by the

formulas 1 f ...

A function /

**analytic**in an annulus a < |z— Zq! < ji has a unique Laurentexpansion /(«) = f «,(«-*)», which converges uniformly and absolutely in every

annulus a+e :£ |z— z0| £S /S — e with e > 0. The coefficients ap are given by the

formulas 1 f ...

Page 230

The largest number n such that a_ nl ^ 0 is called the order of the polo 20. If no a„

with p < 0 is non-zero, and if we put f(z0) = a^, then / becomes

< r, so that the singularity at z = 2q is removable. If ap — 0 for p < 0, z0 is called ...

The largest number n such that a_ nl ^ 0 is called the order of the polo 20. If no a„

with p < 0 is non-zero, and if we put f(z0) = a^, then / becomes

**analytic**in |z — z0l< r, so that the singularity at z = 2q is removable. If ap — 0 for p < 0, z0 is called ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero