Linear Operators, Part 2 |
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Page 1218
... u be a finite positive regular measure on the Borel sets of a topological space R. Then , for every B - space valued u - measur- able function f on R and every & > 0 there is a Borel set o in R with μ ( o ) < ε and such that the ...
... u be a finite positive regular measure on the Borel sets of a topological space R. Then , for every B - space valued u - measur- able function f on R and every & > 0 there is a Borel set o in R with μ ( o ) < ε and such that the ...
Page 1221
... measurable sets , and it follows that Om is u - measurable , completing the proof of statement ( i ) . To complete the proof of the theorem , suppose that the functions W1 ( · , λ ) , . . . , W2 ( · , λ ) are not linearly independent for u ...
... measurable sets , and it follows that Om is u - measurable , completing the proof of statement ( i ) . To complete the proof of the theorem , suppose that the functions W1 ( · , λ ) , . . . , W2 ( · , λ ) are not linearly independent for u ...
Page 1341
... measure u . If { m } is the matrix of densities of p , with respect to up , then there exist non- negative μ - measurable functions q ;, i = 1 , ... , n , μ - integrable over each bounded interval , and μ - measurable functions a¡¡ , 1 ...
... measure u . If { m } is the matrix of densities of p , with respect to up , then there exist non- negative μ - measurable functions q ;, i = 1 , ... , n , μ - integrable over each bounded interval , and μ - measurable functions a¡¡ , 1 ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero