## 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 from ...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(p) = 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(p) = 0 if p is in C. (U. I.). Let K be a compact

subset of ...

Page 1651

and p

so that G(p) = F(prop) = F(q). Thus G|I = F. If KCP = 0 and the function p in Co(IU

Io)

and p

**vanishes**outside K, then pkp-p**vanishes**outside a compact subset of I–Cr,so that G(p) = F(prop) = F(q). Thus G|I = F. If KCP = 0 and the function p in Co(IU

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

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero