## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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If i Sp < 0 , it is readily seen from Corollary III.3.8 and the regularity of 2 that the collection of functions which are

If i Sp < 0 , it is readily seen from Corollary III.3.8 and the regularity of 2 that the collection of functions which are

**continuous**and vanish outside of compact sets is dense in L ( R ) . Hence for f in L , ( R ) let k be such a ...Page 968

PROOF . Verification that the neighborhoods N ( h , K , € ) are a base for a topology will be left to the reader . If heN ( h , K , e ) and hq E N ( ho , K , ε ) then hy h , E N ( hho , K2 , ε ) so that multiplication is

PROOF . Verification that the neighborhoods N ( h , K , € ) are a base for a topology will be left to the reader . If heN ( h , K , e ) and hq E N ( ho , K , ε ) then hy h , E N ( hho , K2 , ε ) so that multiplication is

**continuous**.Page 1903

( See also Absolutely

( See also Absolutely

**continuous**functions ) as a B - space , additional properties , IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6 characterizations of C - space , ( 396397 ) on a compact space ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

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