## Linear Operators: General theory |

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Page 224

Functions of a Complex Variable In some of the chapters to follow , and

especially in Chapter VII , we shall use extensions of certain well - known results

in the theory of

functions ...

Functions of a Complex Variable In some of the chapters to follow , and

especially in Chapter VII , we shall use extensions of certain well - known results

in the theory of

**analytic**functions of a complex variable to the case where thefunctions ...

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 f

ay ( 2 ...

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 f

**analytic**in an annulus « < 12 — 2o ! < B has a unique Laurent expansion f ( x ) =ay ( 2 ...

Page 586

Then there exists a 8 > 0 such that if lul < d , then U C ( T ( u ) ) and R ( 2 ; T ( u ) )

is an

such that if sul < 1 , then Ū CO ( T ( u ) ) . Let 8 s & be chosen such that T ( 0 ) — T

...

Then there exists a 8 > 0 such that if lul < d , then U C ( T ( u ) ) and R ( 2 ; T ( u ) )

is an

**analytic**function of u for each à € U . PROOF . By Lemma 3 , there is a &such that if sul < 1 , then Ū CO ( T ( u ) ) . Let 8 s & be chosen such that T ( 0 ) — T

...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero