## Linear Operators: General theory |

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Page 3

The function f is said to be an extension of the function g and g a restriction of f if

the

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

...

The function f is said to be an extension of the function g and g a restriction of f if

the

**domain**of f contains the**domain**of g , and f ( x ) = g ( x ) for x in the**domain**ofg . The restriction of a function f to a subset A of its

**domain**is sometimes denoted...

Page 230

Let | be an analytic function defined on a connected

plane and having its values in a complex B - space X . Then \ | ( z ) does not have

its maximum at any point of the

Let | be an analytic function defined on a connected

**domain**D in the complexplane 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 y ( z ) is identically constant .Page 538

Nelson Dunford. G . Some miscellaneous convexity inequalities . 48 ( Hadamard

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

a < 2 < b and having values in a B - space X . Show that if M ( r ) = max \ | ( z ) ) ...

Nelson Dunford. G . Some miscellaneous convexity inequalities . 48 ( Hadamard

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

**domain**a < 2 < b and having values in a B - space X . Show that if M ( r ) = max \ | ( z ) ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 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 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