## Linear Operators: Spectral theory |

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Page 966

For some choice of f the integral on the right of [ * ] is not zero and since , by

Lemma 1 ( d ) , the integral on the left of [ * ] is

agrees almost everywhere with a

of ...

For some choice of f the integral on the right of [ * ] is not zero and since , by

Lemma 1 ( d ) , the integral on the left of [ * ] is

**continuous**, we conclude that hmagrees almost everywhere with a

**continuous**function . By redefining hm on a setof ...

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 , EN ( h , K , e ) and h , e N ( ho , K , ε ) then hy h , e N

( hho , K2 , ε ) so that multiplication is

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

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

( hho , K2 , ε ) so that multiplication is

**continuous**. If hy e N ( h , K , ε ) then hil e ...Page 1903

... definition , IV.6.21 ( 276 ) , IX.2.15 ( 872 ) Completeness , weak . ( See Weak

comnon - existence in Lo , 0 < p < 1 , V.7.37 ( 438 )

also Absolutely

15 ...

... definition , IV.6.21 ( 276 ) , IX.2.15 ( 872 ) Completeness , weak . ( See Weak

comnon - existence in Lo , 0 < p < 1 , V.7.37 ( 438 )

**Continuous**functions . ( Seealso Absolutely

**continuous**functions ) as a B - space , additional properties , IV.15 ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

57 other sections not shown

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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 complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure 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