An example of a not locally bounded convex function

Question

It's well known that any convex function $f$ in $\mathbb{R}^d$ is locally bounded (for any $x \in \mathbb{R}^d$ there is an open set $U$ such that $f(U)$ is a bounded set). Are there some nice infinite-dimensional examples where this fails? Could you, for example, take an infinite dimensional separable Hilbert space $\mathcal{H}$ and use its basis to construct an explicit example of a convex function that is not locally bounded?