Convex, closed, symmetric sets as open balls

Question

So we recently saw the way that we can, given a convex symmetric body in $\Bbb R^n$, construct a norm such the closed unit ball is the convex set. We're wondering whether this holds for infinite dimensions. We keep looking at the argument, but it doesn't seem to use the finite dimension of $\Bbb R^n$. Does this hold in infinite dimensions?