It's rather trivial to prove that when a normed vector space X over the real numbers, or the complex numbers, has a Schauder Basis then X is separable. Since you can construct a dense&countable set by using the rational numbers, which you know are countable.
What I'm wondering is, what can we say when X is a vector space over an arbitrary Field F. Can we still find a countable set and then proceed to show that it's also dense? Or is it necessary to take X as a vec space over R/C ?