# Does every inner product fail to turn $C[0,1]$ into a Hilbert space? [duplicate]

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.

• 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 $C[0,1]$. Aug 2, 2014 at 19:20
• 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]. Aug 2, 2014 at 19:21
• @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... Aug 2, 2014 at 19:35
• @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. Aug 2, 2014 at 20:21