Separable vs countable

Question

I want to show that $l^2(X)$ is separable iff $X$ is countable. Note that a space is separable if it has a countable dense subset. I can see that if $X$ is countable, then $l^2(X)$ is separable. To prove the other direction, I need a countable dense set. How can I construct such a set? Any help would be highly appreciated. Thanks so much.

Answer 1

If $X$ is uncountable consider the elmenets $f_x, x \in X$ defined by $f_x(y)=1$ if $y=x$ and $0$ otherwise. Note that $\|f_x-f_y\|=\sqrt 2$ whenever $x \neq y$. The balls $B(f_x,\frac 1 {\sqrt 2})$ form a disjoint uncountable family open sets which implies that the space is not separable.