## Linear Operators, Part 1 |

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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 230

If no a, with p < 0 is non-zero, and if we put f(zo) = an; then f becomes

2–zo & r, so that the singularity at z = zo is removable. If a, - 0 for p < 0, z0 is

called a zero of f; thus zo is a zero of f if f(zo) = 0. If, in this case, a, - 0 for p < n but

a, ...

If no a, with p < 0 is non-zero, and if we put f(zo) = an; then f becomes

**analytic**in2–zo & r, so that the singularity at z = zo is removable. If a, - 0 for p < 0, z0 is

called a zero of f; thus zo is a zero of f if f(zo) = 0. If, in this case, a, - 0 for p < n but

a, ...

Page 586

o Then there evists a 6 - 0 such that if u → 6, then U Co(T(u)) and R(A; T(u)) is an

u → 31, then U Co(T(u)). Let 0 < 0, be chosen such that T(0)—T(u) < inf R(A; ...

o Then there evists a 6 - 0 such that if u → 6, then U Co(T(u)) and R(A; T(u)) is an

**analytic**function of u for each A. e U. PRoof. By Lemma 3, there is a 6, such that ifu → 31, then U Co(T(u)). Let 0 < 0, be chosen such that T(0)—T(u) < inf R(A; ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero