Can any uncountable dimensional real vector space be made into a Banach space?

Question

On any real vector space $V$ of uncountable dimension , can we always define a norm such that endowed with that norm , $V$ becomes a complete normed linear space ? ( I know it can be done if $V$ is finite dimensional but what if $V$ is infinite dimensional ? The only thing I know is that any infinite dimensional Banach space must be of uncountable dimension )

Answer 1

I'm not a set theory expert, but I'd argue as follows:

Begin with your favorite Banach space $B$, forget its metric, and consider only its vector space structure. As a vector space, $B$ possesses a Hamel base $(e_\iota)_{\iota\in I}$ of a certain cardinality $|I|$.

Now if the dimension of your vector space $V$ equals $|I|$ you can copy the Banach space structure of $B$ over to $V$.