TITLE: Is there a vector space that cannot be an inner product space?
QUESTION [47 upvotes]: Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks a lot in advance.

REPLY [8 votes]: Christian Blatter's answer shows that, assuming the axiom of choice, every vector space can be equipped with an inner product.
Without the axiom of choice, this can fail.  As I show in Inner product on $C(\mathbb R)$, it is consistent with ZF+DC that the vector space $C(\mathbb{R})$ of continuous functions on $\mathbb{R}$ does not admit any inner product, nor even any norm.
The idea is that $C(\mathbb{R})$ already has a "nice" topology which is not compatible with an inner product, and under appropriate axioms consistent with ZF+DC, there are "automatic continuity" results saying that it cannot have any other "nice" topology.