Show that $Y $ is separable.

Question

Let $(X,\|.\|) $ be a banach space and $(A_n)_n $ be a famile of separable subset of $X$.

Let $Y $ be a linear subset of $X $ which is generated by the union of $(A_n)_n$.

Show that $Y $ is separable.

An idea please.

Answer 1

Each $A_n$ has a dense denumerable set, say $\{a_k^n: k\in\Bbb N\}$. Take the (numerable) infinite union of all these sequences: $\{y_j:\, j\in\Bbb N\}:=\bigcup_{n\in\Bbb N}\{x_k^n:\, k\in\Bbb N\}$.

Now you can take all (finite) linear combinations of the $(y_j)_j$ with RATIONAL coefficients (this can be denoted by ${\rm span}_{\Bbb Q}\{y_j:\, j\in\Bbb N\}$).

It is clear that this is a denumerable set and it is easy to show (using that $\Bbb Q$ is dense in $\Bbb R$) that this set is dense in $Y$.