Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.

For example, when $r=1/(1+\sqrt2)$, we can pick an orthonormal basis $(x_1,...)$ for our Hilbert space and put the centers of the balls at $(1-r)x_i = \sqrt2/(1+\sqrt2)x_i$ for each $i$. The distance between any two centers is thus $\sqrt2/(1+\sqrt2)\sqrt2 = 2r$, so indeed the balls just kiss each other.

Can we fit any larger balls? What is the critical radius $r_\infty$ such that for $r>r_\infty$ we may only fit finitely many balls of radius $r$, but for smaller $r$ we may fit infinitely many?