I found the following question in an exam and not able to completely crack it. I paste the question as it was there:
Suppose $C[a, b]$ denote the set of all real valued continuous functions on $[a, b]$. Define an inner product on it so that it will become an inner product space having the property that for any sequence $\{f_n\}_{n=1}^{\infty} \subset C[a, b] ~ ||f_n-f_m|| \to 0$ as $m, n \to \infty$ if and only if for some $f, ||f_n-f||\to 0$ as $n \to \infty$.
What I understand that this question is asking for an 'inner product' on $C[a, b]$ so that, a sequence $(f_n)$ is Cauchy iff it converges ! i.e., to define an 'complete inner product' on the vector space $C[a, b]$.
- Now although, $\langle f, g \rangle :=\int_a^bfg$ defines an inner product on $C[a, b]$, it is not complete! So we have to define something else.
- Also another fact is that the (norm)topology induced by such an inner product should not be finer (nor weaker) than $\Vert~ \Vert_{\infty}$-topology, since both are going to be Banach-norm topology on $C[a, b]$. i.e. the new norm should be non comparable to the sup norm.
What would be such an inner product on $C[a, b]$ ? Any remark/suggestion is highly appreciated. Thanks.
In principle, Paul Sinclair's first comment settles the question -- except for the dimension issues. The point is, that all separable Banach spaces are isomorphic as vector spaces because they have the same dimension $c$ (cardinality of $\mathbb R$). There is thus a linear bijective map $T:C[a,b]\to \ell^2$, the Hilbert space of square summable sequences. For two functions $f,g\in C[a,b]$ we can thus define a scalar product $\langle f,g\rangle_{C[a,b]}=\langle T(f),T(g)\rangle_{\ell^2}$. This makes $C[a,b]$ a Hlbert space: For a Cauchy sequence $f_n$ the sequence $T(f_n)$ is Cauchy in $\ell^2$ and hence converges to some $x$ in $\ell^2$ so that $f_n$ converges to $T^{-1}(x)$.
However, this scalar product has nothing to do with the usual topology of $C[a,b]$ of uniform convergence (e.g., because of the closed graph theorem). The existence requires the axiom of choice (to have Hamel bases), and for all concrete questions this existence result is completely useless.