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I have the following question:

With the usual inner products that come in mind, $C[0,1]$ is not a Hilbert space. But is it true that every inner product fails to turn $C[0,1]$ into a Hilbert space? If so, then I would be curious, how to prove this.

Thank you very much for the help!

Yes, I would be interested in the case that the inner product on $C[0,1]$ gives an equivalent topology as the sup norm on $C[0,1]$, but I forgot to clarify this, when typing the question.

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    $\begingroup$ I'd be every surprised if one could explicitly describe an inner product making it a Hilbert space, but since its dimension is $2^{\aleph_0}$, just like the dimension of $\ell^2(\mathbb{N})$, there is an isomorphism between the spaces, and via that you can transport the Hilbert space structure to $C[0,1]$. $\endgroup$ Commented Aug 2, 2014 at 19:20
  • $\begingroup$ If you don't care about the topology, you just have an infinite dimensional vector space. So you probably want to specify that the inner product on C[0,1] gives you an equivalent topology as the sup norm on C[0,1]. $\endgroup$ Commented Aug 2, 2014 at 19:21
  • $\begingroup$ @DanielFischer: Thank you very much for the idea in your comment. Yes, I would be interested in the case that the inner product on $C[0,1]$ gives an equivalent topology as the sup norm on $C[0,1]$, but I forgot to clarify this, when typing the question...sorry... $\endgroup$ Commented Aug 2, 2014 at 19:35
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    $\begingroup$ @MonkeyDRuffy: You can edit your question to include your clarification (use the "edit" button). This will help keep people from getting confused if they don't see your comment. $\endgroup$ Commented Aug 2, 2014 at 20:21

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