## Linear Operators: Spectral theory |

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Page 984

The set of functions f in L1(R) for which f

dense in L1(R). - PROOF. It follows from Lemma 3.6 that the set of all functions in

L2(R, 3, u) which

The set of functions f in L1(R) for which f

**vanishes**in a neighborhood of infinity isdense in L1(R). - PROOF. It follows from Lemma 3.6 that the set of all functions in

L2(R, 3, u) which

**vanish**outside of compact sets is dense in this space, and ...Page 1650

If F

parts of this lemma are left to the reader as an exercise. To prove (v), we must

show from our hypothesis that F(q) = 0 if p is in C. (U. I.). Let K be a compact

subset of ...

If F

**vanishes**in each set I, it**vanishes**in U.I. PRoof. The proofs of the first fourparts of this lemma are left to the reader as an exercise. To prove (v), we must

show from our hypothesis that F(q) = 0 if p is in C. (U. I.). Let K be a compact

subset of ...

Page 1651

and p

so that G(p) = F(pkop) = F(q). Thus G|I = F. If KCF = 0 and the function p in C. (I U

Io)

and p

**vanishes**outside K, then pkp-p**vanishes**outside a compact subset of I–CF,so that G(p) = F(pkop) = F(q). Thus G|I = F. If KCF = 0 and the function p in C. (I U

Io)

**vanishes**outside K, then it is clear that popk**vanishes**outside a compact ...### 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 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 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