Let $K$ be the completion of a field $k$ equipped with a norm $\|\cdot\|$. What can we say about it's cardinality, $|K|$?
By Cantor's completion process we can treat $K$ as a quotient of $c \le k^\infty$ (space of convergent sequences) by $c_0$, space of null sequences. So definitely $|K| \le |k|^{\aleph_0}$. Is the inequality with flipped sign also true?
Sometimes yes ($k = \mathbb R$). Would an assumption that $k = K$ help?