## Linear Operators: Spectral theory |

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Page 984

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

in L2 ( R , B , u ) which vanish outside of compact sets is

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

**dense**in Ly ( R ) . PROOF . It follows from Lemma 3.6 that the set of all functionsin L2 ( R , B , u ) which vanish outside of compact sets is

**dense**in this space ...Page 1246

We may also regard A as a mapping from the

the space Hı . In this case A is still continuous , for | Axli = ( Ax , Ax ) , ( AX , Ax ) , =

( A2x , x ) , ( Ax , x ) , 2 e ( T ) , and , by the inequalities above , ( Ax , x ) Axl x Axix

...

We may also regard A as a mapping from the

**dense**subspace D ( T ) of H intothe space Hı . In this case A is still continuous , for | Axli = ( Ax , Ax ) , ( AX , Ax ) , =

( A2x , x ) , ( Ax , x ) , 2 e ( T ) , and , by the inequalities above , ( Ax , x ) Axl x Axix

...

Page 1905

... rules of , ( 2 )

7.40-41 ( 438–439 )

functions in TM and L , III.9.17 ( 170 ) , IV.8.19 ( 298 )

in Lp ...

... rules of , ( 2 )

**Dense**convex sets , V.7.27 ( 437 )**Dense**linear manifolds , V.7.40-41 ( 438–439 )

**Dense**set , definition , 1.6.11 ( 21 )**density**of continuousfunctions in TM and L , III.9.17 ( 170 ) , IV.8.19 ( 298 )

**density**of simple functionsin Lp ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 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