Let $V$ be a normed space with a norm $\|\cdot\|.$
The polarisation identity Wiki page states that
In a normed space $(V,\|\cdot\|),$ if the parallelogram law holds, then there is an inner product $\langle\cdot,\cdot\rangle$ on $V$ such that $\|x\|^2 = \langle x,x\rangle $ for all $x\in V$.
This post provided a sketch of a proof to the statement above.
I would like to ask whether the induced equation $\|x\|^2 = \langle x,x\rangle$ for inner product is unique, that is,
Question: If a norm satisfies the parallelogram law, then does there exist an inner product $\langle\cdot,\cdot\rangle'$ on $V$ such that $\|x\|^2 \neq \langle x,x\rangle'$ for some $x\in V?$
In other words, if a norm induces an inner product, is the inner product unique in the sense that $\|x\|^2 = \langle x,x\rangle?$