$\exists\, f\in L^2(X)$ s.t. $\{gf\mid g\in C(X, \mathbb{C})\} \subset L^2(X)$ is dense

Asked Jun 14 '23 at 08:00

Active Dec 17 '23 at 17:32

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Let $\mu$ be a positive, $\sigma$-finite Borel measure on a compact space $X$. Can we prove $\exists\, f\in L^2(X, \mu)$ such that $\{gf\mid g\in C(X, \mathbb{C})\} \subset L^2(X, \mu)$ is dense? I am aware that the set of all continuous functions is dense in $L^2(X,\mu)$, but that is probably not useful here.

This problem is originally from Conway's 'A Course in Operator Theory', chapter 1.6, exercise 3. Using this post, I was able to reduce the problem to the above form.

Any help is appreciated.

edited Dec 17 '23 at 17:32

asked Jun 14 '23 at 08:00

Vinay Deshpande

2

What kind of space is $X$? And what is $\mu$? – geetha290krm Jun 14 '23 at 08:05
@geetha290krm Please see edit. There is no specific information about the space $X$ though. – Vinay Deshpande Jun 14 '23 at 08:13
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As far as I can see $X$ is assumed compact in the exercise in Conway's book. Also, do we really want $\mu$ to be any positive Borel measure without any regularity assumptions? – user920957 Dec 14 '23 at 12:57
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@user920957 indeed, with no regularity assumptions on $\mu$ the statement is false even if the space is compact (just take the counting measure on $[0,1]$: every integrable function must have countable support). – Pelota Dec 14 '23 at 13:40
I guess that if the set of all continuous functions is dense in $L^2(X.\mu)$ then we can pick $g\equiv 1$. – Alex Ravsky Dec 14 '23 at 22:32
@AlexRavsky We have to prove "$\exists f$". So we have to pick $f$, not $g$, isn't it? – Vinay Deshpande Dec 15 '23 at 06:20
Oops, sorry. Thanks for noticing. Right, I meant pick $f\equiv 1$ – Alex Ravsky Dec 15 '23 at 06:48
@Pelota I found a similar post. I will try to mimic the proof. https://math.stackexchange.com/q/972902/628415 – Vinay Deshpande Dec 17 '23 at 17:30
@user920957 Thanks to your comment, I realized the author meant to say $\mu$ is $\sigma$-finite. Also, in the post I linked, I did mention $X$ is compact. – Vinay Deshpande Dec 17 '23 at 17:30

$\exists\, f\in L^2(X)$ s.t. $\{gf\mid g\in C(X, \mathbb{C})\} \subset L^2(X)$ is dense

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