## 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 228

Let Vbe a bounded open subset of U whose closure is contained in U. If fn

converges at each point of U to a function f on U then f is

partial derivatives of /„ of arbitrarily large order converge to the corresponding

partial ...

Let Vbe a bounded open subset of U whose closure is contained in U. If fn

converges at each point of U to a function f on U then f is

**analytic**in U, and thepartial derivatives of /„ of arbitrarily large order converge to the corresponding

partial ...

Page 230

The largest number n such that a_]n] ^ 0 is called the order of the pole z0. If no

a p with p < 0 is non-zero, and if we put f(z0) = a0. then / becomes

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

called ...

The largest number n such that a_]n] ^ 0 is called the order of the pole z0. If no

a p with p < 0 is non-zero, and if we put f(z0) = a0. then / becomes

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

called ...

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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 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