- #1
D_Miller
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Technically this isn't homework, but just something I saw another user state without proof in a very different thread. I believe, however, that it is specific enough to pass as a "homework question" so I thought I'd pretend that it was and post it here, because I'm getting a bit frustrated with it:
Let f be a continuous, complex-valued function on [0, 1] and define a linear operator [tex]M_f[/tex] on [tex]L^2[0,1][/tex] by [tex](M_f g)(x)=f(x)g(x)[/tex]. Then
(a) [tex]M_f[/tex] is bounded and [tex]||M_f||=||f||_{\infty}[/tex].
(b) Suppose that [tex]M_f[/tex] is one-to-one. Then the range of [tex]M_f[/tex] is closed if and only if [tex]f(x)=0[/tex] for all x∈[0,1]. In that case [tex]M_f[/tex] is one-to-one and
onto, and the inverse map is bounded.
Part (a) is rather trivial, but part (b) bugs me. I have never seen this result before, and I cannot seem to prove it. My original idea was to use the open mapping theorem in relation to the kernel of the operator, but I couldn't make it work. I still think the open mapping theorem or a similar result is the right way forward, but I would very much appreciate it if someone could write out a proof.
Oh, and if it could be of help to anyone, the injectivity of [tex]M_f[/tex] follows when the set of zeroes of f has measure 0.
Let f be a continuous, complex-valued function on [0, 1] and define a linear operator [tex]M_f[/tex] on [tex]L^2[0,1][/tex] by [tex](M_f g)(x)=f(x)g(x)[/tex]. Then
(a) [tex]M_f[/tex] is bounded and [tex]||M_f||=||f||_{\infty}[/tex].
(b) Suppose that [tex]M_f[/tex] is one-to-one. Then the range of [tex]M_f[/tex] is closed if and only if [tex]f(x)=0[/tex] for all x∈[0,1]. In that case [tex]M_f[/tex] is one-to-one and
onto, and the inverse map is bounded.
Part (a) is rather trivial, but part (b) bugs me. I have never seen this result before, and I cannot seem to prove it. My original idea was to use the open mapping theorem in relation to the kernel of the operator, but I couldn't make it work. I still think the open mapping theorem or a similar result is the right way forward, but I would very much appreciate it if someone could write out a proof.
Oh, and if it could be of help to anyone, the injectivity of [tex]M_f[/tex] follows when the set of zeroes of f has measure 0.
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