Using the axiom of choice one can show that for each ($\mathbb{R}$-) vector space $V$ there exists a function $\|\cdot\| : V \rightarrow \mathbb{R}$ so that $(V,\|\cdot\|)$ is a normed vector space.
So I wondered: Does there also for each ($\mathbb{R}$-) vector space $V$ exist a norm $\|\cdot\|$ so that $(V, \|\cdot\|)$ is a banach space? If yes, why? If not, can we construct a counter example?
I would love to add a “what I have already tried” section here, but sadly I do not know what to try. (I just feel this should be false and therefore hope for a nice counter example.)
No. The baire category theorem guarantees that no normed space with dimension equal to the cardinality of the natural numbers can be a Banach space.