Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.

Question

Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of the inner product is the same as $\tau_1$ ?

I suspect the answer is no. A sufficent condition for that to happen is that the parallelogram law holds. Is there a weaker known sufficient condition ?

Thank you