## 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

on G and with values in X is said to be

partial ...

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

complex valued

**analytic**functions of one complex variable ... A function ; definedon G and with values in X is said to be

**analytic**on G if f is continuous and the firstpartial ...

Page 228

... by 21 Zn is

closure of the bounded open set U which is bounded by a finite collection B of

closed disjoint rectifiable Jordan curves oriented in the customary positive sense

...

... by 21 Zn is

**analytic**in U , X ... Un - 1 : If f is**analytic**in a neighborhood of theclosure of the bounded open set U which is bounded by a finite collection B of

closed disjoint rectifiable Jordan curves oriented in the customary positive sense

...

Page 229

These facts , as well as the following remarks about Laurent series may all be

proved by the standard arguments used for complex functions . A function

—20 ) " ...

These facts , as well as the following remarks about Laurent series may all be

proved by the standard arguments used for complex functions . A function

**analytic**in an annulus . < 2-20 < B has a unique Laurent expansion f ( x ) E ap ( 2—20 ) " ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero