## Linear Operators, Part 1 |

### From inside the book

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Page 3

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

|| A . If 1 : A → B , and for each bef ( A ) there is only one a € À 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 by|| A . If 1 : A → B , and for each bef ( A ) there is only one a € À 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 ( 2 ) is identically constant .Page 605

2 In the spaces L , ( - 1 , 1 ) 1 Spoo on the unit circle , let T be the differentiation

operator ( Tx ) ( t ) = x ' ( t ) with

and x'EL ( -- , ) } . Show that T is a closed unbounded operator whose

2 In the spaces L , ( - 1 , 1 ) 1 Spoo on the unit circle , let T be the differentiation

operator ( Tx ) ( t ) = x ' ( t ) with

**domain**D ( T ) = { x | x is absolutely continuousand x'EL ( -- , ) } . Show that T is a closed unbounded operator whose

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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