## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 910

This

This

**shows**that U , preserves inner products and is thus one - to - one and continuous . It therefore has a unique extension U from Di = to Hj the Ly - closure of the set of bounded Borel functions , i.e. , to Lg ( u ) .Page 985

It follows from Corollary II.3.13 that f * g is in l for every g in Ly ( R ) , which

It follows from Corollary II.3.13 that f * g is in l for every g in Ly ( R ) , which

**shows**that L is an ideal and thus L 2 I. Conversely let | be in the closed ideal f in L ( R ) and suppose that the bounded linear functional F vanishes ...Page 987

Q.E.D. The result just proved

Q.E.D. The result just proved

**shows**that if the bounded measurable function 9 on R is not zero almost everywhere there is at least one character of R in the Ly - closed linear manifold I ( 9 ) of L ( R ) which is determined by the ...### What people are saying - Write a review

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additive adjoint operator algebra analytic assume B-algebra 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 eigenvalues element equal equation Exercise exists extension fact finite dimensional follows formal formal differential operator formula function function f given Hence Hilbert space Hilbert-Schmidt ideal identity independent inequality integral interval isometric isomorphism Lemma limit linear matrix measure multiplicity neighborhood norm normal operator obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solution spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero