## Linear Operators: General theory |

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Page 224

Functions of a Complex Variable In some of the chapters to follow, and especially

in Chapter VII, we shall use extensions of certain well-known results in the theory

of

Functions of a Complex Variable In some of the chapters to follow, and especially

in Chapter VII, we shall use extensions of certain well-known results in the theory

of

**analytic**functions of a complex variable to the case where the functions are ...Page 229

A function /

X /(*) = 2 a^z-z^Y, P=— OB which converges uniformly and absolutely in every

annulus a+e :g |z— Zq\ ^ /3 — e with e > 0. The coefficients a„ are given by the ...

A function /

**analytic**in an annulus a < — z0| < /? has a unique Laurent expansionX /(*) = 2 a^z-z^Y, P=— OB which converges uniformly and absolutely in every

annulus a+e :g |z— Zq\ ^ /3 — e with e > 0. The coefficients a„ are given by the ...

Page 230

If no a„ with p < 0 is non-zero, and if we put /(z0) = a0. then / becomes

z— z„| < r. so that the singularity at z = Zj, is removable. If av = 0 for p < 0, z0 is

called a zero of /; thus z0 is a zero of / if /(z0) = 0. If, in this case, aB = 0 for p < n

but ...

If no a„ with p < 0 is non-zero, and if we put /(z0) = a0. then / becomes

**analytic**in |z— z„| < r. so that the singularity at z = Zj, is removable. If av = 0 for p < 0, z0 is

called a zero of /; thus z0 is a zero of / if /(z0) = 0. If, in this case, aB = 0 for p < n

but ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

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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