Suppose that $X$ is a separable Banach space. Then there exists a sequence $\{ x_n: n \in \mathbb{N} \}$ such that the sequence is dense in a unit ball $B_X$.
Question: Whenever we talk about separable Banach space, I notice that we always have a sequence dense in a unit ball. Why we talk about unit ball instead of the whole space $X$? Is it because all $x \in X$ can be normalised?
Best Answer
Two reasons: if $X$ is a metric space (as a Banach space is) and $X$ is separable (i.e. has a countable dense subset), then every subset of $X$ also has a countable dense subset. This holds because having a countable dense subset and having a countable base (for the topology) are equivalent in metric spaces. So $X$ separable implies $B_X$ separable by being metric alone.
But because of the Banach space structure, the reverse also holds: if $B_X$ has a countable dense subset $\{x_n: n \in \mathbb{N}\}$, then the set $\{m\cdot x_n: m,n \in \mathbb{N} \}$ is dense in $X$:
Let $x \in X$, $r>0$. Pick $m \in \mathbb{N}$ with $m > ||x||$. Then $\frac{1}{m} \cdot x \in B_X$, so we pick $x_n$ within distance $\frac{r}{m}$ of this point, by denseness. Then $||m \cdot x_n - x || = m||x_n - \frac{1}{m} \cdot x|| < m \frac{r}{m} = r$ as required.
And so $X$ also has a countable dense subset. And because all balls are homeomorphic, the unit ball isn't even special in that regard (if one open ball is separable, they all are).
(Note trivially that in a discrete metric space any unit ball is trivial, so separable, but only countable discrete metric spaces are separable, so in general we cannot go from separability of a ball to that of the whole space.)