Let $X$ be a real normed space and $k \ge 1$, and give $\mathcal{G}_k X$ the Hausdorff distance between unit spheres. If $X$ is complete then so is $\mathcal{G}_k X$, and if $X$ is separable then so is $\mathcal{G}_k X$. In particular $\mathcal{G}_k X$ is Polish when $X$ is a separable Banach space. Completeness of spaces of closed subspaces under the gap distance is classical (Kato 2013, Chapter IV, §2.1).