## Linear Operators: Spectral theory |

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Page 1650

which y vanishes . ... complement in I of the largest open set in I in which F

vanishes , i . e . , which is the complement in I of the union of all the open

of ...

**subsets**of I and let F be in D ( I ) . ... Let K be a compact**subset**of Ugla outside ofwhich y vanishes . ... complement in I of the largest open set in I in which F

vanishes , i . e . , which is the complement in I of the union of all the open

**subsets**of ...

Page 1660

Of course , if we deal only with

identifications are made and the phrases “ open set , ” “ closed ... If I is an open

02 .

Of course , if we deal only with

**subsets**of the interior of the cube C in E " , noidentifications are made and the phrases “ open set , ” “ closed ... If I is an open

**subset**of C , in the modified sense explained in the preceding paragraph , then02 .

Page 1695

Lemma 13 and the following lemma taken together give considerable insight into

the nature of distributions in general . 15 LEMMA . Let F be a distribution in the

open

...

Lemma 13 and the following lemma taken together give considerable insight into

the nature of distributions in general . 15 LEMMA . Let F be a distribution in the

open

**subset**1 of En . Let { In } be a sequence of open**subsets**of I whose union is...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero