## 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 ...

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 thefunctions ...

Page 228

... sequence of vector valued functions each defined and

U in the space of the complex variables ...

derivatives of in of arbitrarily large order converge to the corresponding partial

derivatives ...

... sequence of vector valued functions each defined and

**analytic**on an open setU in the space of the complex variables ...

**analytic**in U , and the partialderivatives of in of arbitrarily large order converge to the corresponding partial

derivatives ...

Page 230

Nelson Dunford, Jacob T. Schwartz. The largest number n such that a _ ini + 0 is

called the order of the pole Zo . If no a , with p < 0 is non - zero , and if we put f (

20 ) = do , then f becomes

is ...

Nelson Dunford, Jacob T. Schwartz. The largest number n such that a _ ini + 0 is

called the order of the pole Zo . If no a , with p < 0 is non - zero , and if we put f (

20 ) = do , then f becomes

**analytic**in 3 — 2o < r , so that the singularity at z = zois ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero