## Linear Operators: General theory |

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The restriction of a function f to a subset A of its

The restriction of a function f to a subset A of its

**domain**is sometimes denoted by f | A . If f : A → B , and for each bef ( A ) there is only one a € A with f ( a ) = b , f is said to have an inverse or to be one - to - one .Page 230

Let | be an analytic function defined on a connected

Let | be an analytic function defined on a connected

**domain**D in the complex plane and having its values in a complex B - space X. Then | ( z ) does not have its maximum at any point of the**domain**D , unless \ | ( 2 ) is identically ...Page 538

48 ( Hadamard three circles theorem ) Let f be an analytic function defined in the annular

48 ( Hadamard three circles theorem ) Let f be an analytic function defined in the annular

**domain**a < 121 < b and having values in a B - space X. Show that if M ( r ) = max \ | ( z ) ) , log M ( r ) is a convex function of log r , a < r ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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