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, and other answers to the same question, for some interesting points related to this).
Best Answer
Let me try a possible answer. Take a model of $ZF$ where the axiom of choice for a denumerable family of finite sets holds but where there is an infinite Dedekind finite set $B$ (this model can be checked to exist, for instance, here; $\mathcal{M}32$ is one such model). Then $\ell_2(B)$ is an infinite dimensional Hilbert space with a Dedekind finite orthonormal base, whose unit ball is, by theorem 2 of the previously cited article, sequentially compact.