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

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Page 961

7 LEMMA . If f is in L ( R ) , L ( R ) and if e is a Borel set in M whose closure does

not contain poo , then S. ( tt ) ( m ) u ( dm ) = 0 [ E ( e ) } ] ( t , y ( e ) ) where y ( e ) is

the vector in Ly ( R ) defined in Lemma 6 . 4

7 LEMMA . If f is in L ( R ) , L ( R ) and if e is a Borel set in M whose closure does

not contain poo , then S. ( tt ) ( m ) u ( dm ) = 0 [ E ( e ) } ] ( t , y ( e ) ) where y ( e ) is

the vector in Ly ( R ) defined in Lemma 6 . 4

**PROOF**. If we write y for yle ) , then ...Page 1459

present corollary follows from Corollary 7 and Definition 25 ( b ) . Q.E.D. 31

COROLLARY . Suppose in addition to the hypotheses of Theorem 8 that the

coefficients Pre ...

**PROOF**. It is obvious from Definition 20 that t is bounded below . Thus thepresent corollary follows from Corollary 7 and Definition 25 ( b ) . Q.E.D. 31

COROLLARY . Suppose in addition to the hypotheses of Theorem 8 that the

coefficients Pre ...

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We shall see , however , that this fact is needed in the course of the

Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

We shall see , however , that this fact is needed in the course of the

**proof**ofTheorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

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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

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