## Linear Operators: Spectral theory |

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Page 910

This

continuous . It therefore has a unique extension U from Di = H to the L , - closure

of the set of bounded Borel functions , i . e . , to L , ( u ) . An elementary continuity

...

This

**shows**that U , preserves inner products and is thus one - to - one andcontinuous . It therefore has a unique extension U from Di = H to the L , - closure

of the set of bounded Borel functions , i . e . , to L , ( u ) . An elementary continuity

...

Page 985

Let F be a continuous linear functional on L ( R ) which vanishes on L , let o be

the bounded measurable function representing F ( IV . 8 . 5 ) , and let f be any

point in L . Then , since 0 = F ) , = SRP ( x ) / ( x − y ) dx equation ( i )

...

Let F be a continuous linear functional on L ( R ) which vanishes on L , let o be

the bounded measurable function representing F ( IV . 8 . 5 ) , and let f be any

point in L . Then , since 0 = F ) , = SRP ( x ) / ( x − y ) dx equation ( i )

**shows**that F...

Page 987

The preceding theorem

every h in L . Furthermore , since t is closed in the L ( R ) topology of L ( R ) , it

follows from Corollary V . 3 . 12 that A is the conjugate - orthogonal complement

of L ...

The preceding theorem

**shows**that there is a point mo in Ř with h ( mo ) = 0 forevery h in L . Furthermore , since t is closed in the L ( R ) topology of L ( R ) , it

follows from Corollary V . 3 . 12 that A is the conjugate - orthogonal complement

of L ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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