Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 952
... continuous and vanish outside of compact sets is dense in L ( R ) . Hence for f in L ( R ) let k be such a continuous function with f - kε . Since k is uniformly continuous , we see , for z sufficiently close to y , that k2 - k ,, < ɛ ...
... continuous and vanish outside of compact sets is dense in L ( R ) . Hence for f in L ( R ) let k be such a continuous function with f - kε . Since k is uniformly continuous , we see , for z sufficiently close to y , that k2 - k ,, < ɛ ...
Page 968
... continuous . If h1 = N ( h , K , e ) then h1 € N ( h − 1 , K , ɛ ) , so the mapping hh - 1 is also continuous . Q.E.D. 15 2 THEOREM . The one - to - one mapping m → hm , whose existence was established in Theorem 11 , is a ...
... continuous . If h1 = N ( h , K , e ) then h1 € N ( h − 1 , K , ɛ ) , so the mapping hh - 1 is also continuous . Q.E.D. 15 2 THEOREM . The one - to - one mapping m → hm , whose existence was established in Theorem 11 , is a ...
Page 1903
... Continuous functions . ( See also Abso- lutely continuous functions ) as a B - space , additional properties , IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6 characterizations of C - space , ( 396 ...
... Continuous functions . ( See also Abso- lutely continuous functions ) as a B - space , additional properties , IV.15 definition , IV.2.14 ( 240 ) remarks concerning , ( 373-386 ) study of , IV.6 characterizations of C - space , ( 396 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero