Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed interval $[a,b]$ with the metric $d$ defined as follows:
$$ d(x,y) \colon= \sup_{a\leq t \leq b} \ |x(t) – y(t)|.$$
Then how to determine whether or not this space is separable?
By definition, a metric space $X$ is said to be separable if it has a countable dense subset, that is, if there is a countable subset $M$ of $X$ such that $\bar{M} = X$.
Best Answer
The space in question is not separable, because you do not have any continuity assumption on the elements. For example, $C([a,b])$ is separable.
To see that your space is not separable, it suffices to construct an uncountable family $(f_i)_i$ in $B([a,b])$ such that $d(f_i, f_j) \geq 1$ for all $i \neq j$ (show this!!, let $(g_n)_n$ be dense in $B([a,b])$, take $\varepsilon = 1/2$ and note that for each $i$ there is some $n_i$ such that $d(f_i, g_{n_i}) < 1/2$. Why does this help you?).
To construct such a family, think for a few minutes or consider the spoiler below.
I leave it to you to check that actually $d(f_x, f_y) = 1$ holds for all $x \neq y$.