Linear Operators, Part 2 |
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Page 952
... continuous and vanish outside of compact sets is dense in L „ ( R ) . Hence for f in L ( R ) let k be such a continuous function with \ ƒ — k | „ < ɛ . Since k is uniformly continuous , we see , for z sufficiently close to y , that k2 ...
... continuous and vanish outside of compact sets is dense in L „ ( R ) . Hence for f in L ( R ) let k be such a continuous function with \ ƒ — k | „ < ɛ . Since k is uniformly continuous , we see , for z sufficiently close to y , that k2 ...
Page 966
... continuous , we conclude that h agrees almost everywhere with a continuous function . By redefining h , on a set of measure zero , we may take it to be continuous . A change of variables in [ * ] shows that for every f in L1 ( R ) , = m ...
... continuous , we conclude that h agrees almost everywhere with a continuous function . By redefining h , on a set of measure zero , we may take it to be continuous . A change of variables in [ * ] shows that for every f in L1 ( R ) , = m ...
Page 968
... continuous . If h1 = N ( h , K , ɛ ) then h11 € N ( h − 1 , K , ɛ ) , so the mapping hh - 1 is also continuous . Q.E.D. 2 15 THEOREM . The one - to - one mapping m → hm , whose existence was established in Theorem 11 , is a ...
... continuous . If h1 = N ( h , K , ɛ ) then h11 € N ( h − 1 , K , ɛ ) , so the mapping hh - 1 is also continuous . Q.E.D. 2 15 THEOREM . The one - to - one mapping m → hm , whose existence was established in Theorem 11 , is a ...
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