Linear Operators, Part 2 |
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Page 873
... set in M of all M , with 2 e A. To see that MA is dense in M the contrary and let suppose { M || x , ( M ) —x ̧ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that & maps open sets onto open sets б ...
... set in M of all M , with 2 e A. To see that MA is dense in M the contrary and let suppose { M || x , ( M ) —x ̧ ( Mo ) ... open . Thus & is continuous . To see that 8-1 is continuous , i.e. , to see that & maps open sets onto open sets б ...
Page 1151
... sets in R. We select an open set G1 in R such that FOKCG , G1 F2 = $ , and then choose an open set H1 such that Φ , F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = Þ . By induction , choose open sets G and H , such that Ğ1 н n n ( F2H1 ... UH - 1 ) = 4 ...
... sets in R. We select an open set G1 in R such that FOKCG , G1 F2 = $ , and then choose an open set H1 such that Φ , F2OK1CH1 , H1 ~ ( F1 ~ Ğ1 ) = Þ . By induction , choose open sets G and H , such that Ğ1 н n n ( F2H1 ... UH - 1 ) = 4 ...
Page 1660
... open set , " " closed set , " etc. , in this slightly modified sense . Since we deal only with multiply periodic functions . throughout , all our functions will be well - defined on the set C even after the indicated identifications are ...
... open set , " " closed set , " etc. , in this slightly modified sense . Since we deal only with multiply periodic functions . throughout , all our functions will be well - defined on the set C even after the indicated identifications are ...
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 domain 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 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 subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero