Linear Operators, Part 2 |
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Page 1048
... o - field of Borel subsets of E is the product o - field of the o - field BR of Borel subsets of R and the o - field Bs of Borel subsets of S , in the sense of Definition III.11.3 . We have -n Sg „ f ( tx ) dx = \ t \ − ” √ g „ f ( x ) ...
... o - field of Borel subsets of E is the product o - field of the o - field BR of Borel subsets of R and the o - field Bs of Borel subsets of S , in the sense of Definition III.11.3 . We have -n Sg „ f ( tx ) dx = \ t \ − ” √ g „ f ( x ) ...
Page 1153
... field ( 2 ) . It is natural to ex- pect that the product measure λ × λ coincides , up to a constant multi- ple ... o - field of Borel sets . Then if f is λ - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2x2 ...
... field ( 2 ) . It is natural to ex- pect that the product measure λ × λ coincides , up to a constant multi- ple ... o - field of Borel sets . Then if f is λ - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2x2 ...
Page 1919
... of , II.4.31-54 ( 74–78 ) Set ( s ) , Borel , III.5.10 ( 137 ) convergence of , ( 126-127 ) , III.9.48 ( 174 ) field of , III.1.3 ( 96 ) 2 - set , III.5.1 ( 133 ) Lebesgue , III.12.9 ( 218 ) open . ( See Open ) o - field of , III.4.2 ...
... of , II.4.31-54 ( 74–78 ) Set ( s ) , Borel , III.5.10 ( 137 ) convergence of , ( 126-127 ) , III.9.48 ( 174 ) field of , III.1.3 ( 96 ) 2 - set , III.5.1 ( 133 ) Lebesgue , III.12.9 ( 218 ) open . ( See Open ) o - field of , III.4.2 ...
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