## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 952

... which are

Hence for f in Ly ( R ) let k be such a

is uniformly

... which are

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

**continuous**function with It - klo < € . Since kis uniformly

**continuous**, we see , for z sufficiently close to y , that \ k , -kelp < € ...Page 968

PROOF . Verification that the neighborhoods N ( h , K , E ) are a base for a

topology will be left to the reader . If h , e N ( h , K , e ) and h , e N ( ho , K , ε ) then

hy h , e N ( hh ,, K2 , ε ) so that multiplication is

then hī ...

PROOF . Verification that the neighborhoods N ( h , K , E ) are a base for a

topology will be left to the reader . If h , e N ( h , K , e ) and h , e N ( ho , K , ε ) then

hy h , e N ( hh ,, K2 , ε ) so that multiplication is

**continuous**. If hy e N ( h , K , E )then hī ...

Page 1903

( See also Absolutely

, IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6

characterizations of C - space , ( 396397 ) on a compact space , 1.5.8 ( 18 ) ...

( 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 , 1.5.8 ( 18 ) ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

### Common terms and phrases

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 Nauk 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