Let $(E,\lVert\cdot\rVert)$ denote a normed vector space. Recall that an inner product space $E$ is a NVS with an additional gadget, namely an inner product that induces the norm. But, a NVS space $E$ is an IPS if and only if the norm satisfies the parallelogram law, i.e. if for all $x,y\in E$, $$\lVert x+y\rVert^2 + \lVert x-y\rVert^2 = 2\lVert x\rVert^2 + 2\lVert y\rVert^2$$ which allows us to define an inner product by letting
$$\langle x,y\rangle = \frac12\left(\lVert x+y\rVert^2-\lVert x\rVert^2 - \lVert y\rVert^2\right).$$
Now, this tells us that the inner product is uniquely determined by the NVS structure. 

Can we take this further? Can we say that the IPS structure on $E$ is uniquely defined by the structure of $E$ as a topological vector space? Or do there exist non isomorphic IPS's that are isomorphic as topological vector spaces?