How to know when the topology of the vector space comes from an inner product?

Question

There is a result which gives an if and only if criterion for knowing when the norm of a normed linear space comes from an inner product; it is so if and only if it satisfies the parallelogram law: \begin{align*} ||x+y||^2 + ||x-y||^2 = 2 ||x||^2 + 2||y||^2 \end{align*} However, this speaks about the norm function. Is there a criterion to address the underlying topology instead? More precisely, given a normed linear space $(V,||.||)$, how do we know that there does or doesn't exist an inner product $\left<.,.\right>$ such that the topology from $(V,\sqrt{\left<.,.\right>})$ matches the topology of the original space? For example, in the case of $(\mathbb{R}^n, ||.||_p)$ the answer to this question is positive for any $p$ since the norms are equivalent in a finite dimensional vector space. What about $(L^p [0,1],||.||_p)$ for instance for $1\leq p \leq \infty$?