Linear Operators, Part 2 |
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Page 1669
... mapping → o M is a continuous mapping of C∞ ( I2 ) into C∞ ( I2 ) . ( See Section 2 for a definition of the topology in these spaces . ) By ( a ) , q → 0 M maps Co ( I ) into Co ° ( I1 ) . By ( a ) again , all the functions of the ...
... mapping → o M is a continuous mapping of C∞ ( I2 ) into C∞ ( I2 ) . ( See Section 2 for a definition of the topology in these spaces . ) By ( a ) , q → 0 M maps Co ( I ) into Co ° ( I1 ) . By ( a ) again , all the functions of the ...
Page 1671
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces Ho , Ao , etc. , under the ...
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces Ho , Ao , etc. , under the ...
Page 1736
... mapping g → §g | C is a continuous mapping of HP ) ( e - 1I ) into HP ( C ) by Lemmas 3.22 and 3.23 , and evidently maps Co ( I ) into Co ( C ) . It follows from Definition 3.15 that it maps HP ( 11 ) into HP ( C ) . Thus , fe ( fo S1 ) ...
... mapping g → §g | C is a continuous mapping of HP ) ( e - 1I ) into HP ( C ) by Lemmas 3.22 and 3.23 , and evidently maps Co ( I ) into Co ( C ) . It follows from Definition 3.15 that it maps HP ( 11 ) into HP ( C ) . Thus , fe ( fo S1 ) ...
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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