## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1962

We next examine the nature of the conjugate space ( HP ) * . A

* on Hp determines p

* x , = x * H ; 1X1 , X ; € H , and , conversely , any set x * , . . . , * * of p linear ...

We next examine the nature of the conjugate space ( HP ) * . A

**linear functional**x* on Hp determines p

**linear functionals**2 * , . . . , on H according to the relations ** x , = x * H ; 1X1 , X ; € H , and , conversely , any set x * , . . . , * * of p linear ...

Page 2066

For other h we fix a function s in Lų for which h ( f ) # 0 and , in terms of the

translation fé ( ) = f ( s — t ) , define the complex valued ... Now let x * be the

continuous

every g in Ly .

For other h we fix a function s in Lų for which h ( f ) # 0 and , in terms of the

translation fé ( ) = f ( s — t ) , define the complex valued ... Now let x * be the

continuous

**linear functional**on L , determined by the relation h ( g ) = x * g forevery g in Ly .

Page 2205

PROOF . By Corollary 10 , B is the range of a countably additive spectral

measure E defined on a o - field E . In view of Corollary 11 and the Hahn -

Banach theorem , it may be assumed that X = sp { EX , Ee B } . For every

PROOF . By Corollary 10 , B is the range of a countably additive spectral

measure E defined on a o - field E . In view of Corollary 11 and the Hahn -

Banach theorem , it may be assumed that X = sp { EX , Ee B } . For every

**linear****functional**y * in ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero