## Linear Operators: Spectral theory |

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Page 910

( z This

( z This

**shows**that U , preserves inner products and is thus one - to - one and continuous . It therefore has a unique extension U ... An elementary continuity argument**shows**that U is an isometric isomorphism between V and L , ( u ) .Page 987

The preceding theorem

The preceding theorem

**shows**that there is a point mo in Ân with him . ... mo ] dx for every h in L which**shows**that the character [ ' , mo ] is in R. Q.E.D. The result just proved**shows**that if the bounded measurable function 9 on R is ...Page 998

MISCELLANEOUS APPLICATIONS XI.4.24 This

MISCELLANEOUS APPLICATIONS XI.4.24 This

**shows**that of vanishes on U , for if m is in U then m - m , is in the complement of V. According to Theorem 13 , y is the limit in the Ly - topology of L- ( R ) of a generalized sequence { 9a } in ...### What people are saying - Write a review

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### Contents

8 | 876 |

859 | 885 |

extensive presentation of applications of the spectral theorem | 911 |

Copyright | |

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### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero