## Linear Operators: Spectral theory |

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Page 984

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

(R, 3, u) which vanish outside of compact sets is

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

**dense**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 1188

If the domain Ø(T) of the operator T is

by definition, of all y in $5 for which (Tw, y) is continuous for a in Ø(T). Since 3)(T)

is

If the domain Ø(T) of the operator T is

**dense**in S) then the domain Ø(T*) consists,by definition, of all y in $5 for which (Tw, y) is continuous for a in Ø(T). Since 3)(T)

is

**dense**in § there is (IV.4.5) a uniquely determined point y” in § such that (Tr, ...Page 1905

... definition, XII.4.9 (1226) De Morgan, rules of, (2)

437)

21)

... definition, XII.4.9 (1226) De Morgan, rules of, (2)

**Dense**convex sets, V.7.27 (437)

**Dense**linear manifolds, V.7.40–41 (438–439)**Dense**set, definition, I.6.11 (21)

**density**of continuous functions in TM and L, III.9.17 (170), IV.8.19 (298)**density**...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad 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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero