Linear Operators, Part 2 |
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Page 868
... field . If I is maximal then X / contains no non - trivial ideals and hence is a field . The desired conclusion follows from Lemma 1.13 and Theorem 1.6 . Q.E.D. Let M be the set of all maximal ideals in the commutative B - algebra X. It ...
... field . If I is maximal then X / contains no non - trivial ideals and hence is a field . The desired conclusion follows from Lemma 1.13 and Theorem 1.6 . Q.E.D. Let M be the set of all maximal ideals in the commutative B - algebra X. It ...
Page 1048
... 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 S n ( 1 ) En √gn f ( tx ) dx -n = En \ t \ − " √ μn f ( x ) dx , t +0 , fe L1 ( E " ) . Thus ...
... 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 S n ( 1 ) En √gn f ( tx ) dx -n = En \ t \ − " √ μn f ( x ) dx , t +0 , fe L1 ( E " ) . Thus ...
Page 1153
... field of Borel sets . Then if f is 2 - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2 × λ measurable . = PROOF . Let ExΣ be the product o - field of subsets of R × R and let E be an open subset of R. Then ...
... field of Borel sets . Then if f is 2 - measurable , the function g defined by g ( x , y ) = f ( x - y ) is 2 × λ measurable . = PROOF . Let ExΣ be the product o - field of subsets of R × R and let E be an open subset of R. Then ...
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BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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