## Linear Operators: General theory |

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Page 3

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

f | A . If 7 : 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 . The inverse function has

...

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

**domain**is sometimes denoted byf | A . If 7 : 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 . The inverse function has

**domain**f...

Page 230

Let f be an analytic function defined on a connected

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

maximum at any point of the

Let f be an analytic function defined on a connected

**domain**D in the complexplane and having its values in a complex B - space X. Then | ( ) does not have its

maximum at any point of the

**domain**D , unless f ( z ) is identically constant .Page 538

Nelson Dunford, Jacob T. Schwartz, William G. Bade. G. Some miscellaneous

convexity inequalities . ( Hadamard three circles theorem ) Let f be an analytic

function defined in the annular

X.

Nelson Dunford, Jacob T. Schwartz, William G. Bade. G. Some miscellaneous

convexity inequalities . ( Hadamard three circles theorem ) Let f be an analytic

function defined in the annular

**domain**a < x < b and having values in a B - spaceX.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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 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