A popular pair of exercises in first courses on functional analysis prove the following theorem:

> The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.

My question is, is the "only if" part of this (i.e., that the unit ball of an infinite-dimensional Banach space is noncompact) necessarily true without some form of the axiom of choice?  The usual proof uses the Hahn--Banach theorem, which may reasonably be regarded as a weak form of the axiom of choice (see [this answer][1], and other answers to the same question, for some interesting points related to this).  


  [1]: https://mathoverflow.net/questions/5351/whats-an-example-of-a-space-that-needs-the-hahn-banach-theorem/5566#5566