Separability of linear span of separable set. Let $(X,d)$ be a separable metric space.
Define a collection of mappings $B:=\{f_x:X\to \mathbb{R}:x\in X\}$ and let $H^{pre}=\mathrm{span}(B)$ denote the linear span of these mappings.
Now equip $H^{pre}$ with an inner product.
Assume that I can find a continuous mapping $g:X\to H^{pre}$, such that $g(X)=B$. By the separability of $X$ it has a countable dense subset $D$ and by continuity we have that $g(D)$ is a countable dense subset of $B$ ( when $B$ is equipped with the restricted metric of $H^{pre}$). 
But how do I show that $\mathrm{span}(B)=H^{pre}$ is separable?
 A: Let $E$ a topological vector space over $\mathbb{R}$ or $\mathbb{C}$, and $S \subset E$. Then $\operatorname{span}_{\mathbb{Q}} S$ is dense in $\operatorname{span}_{\mathbb{R}} S$, and if we have a complex vector space, $\operatorname{span}_{\mathbb{Q}(i)} S$ is dense in $\operatorname{span}_{\mathbb{C}} S$. These follow immediately from the denseness of $\mathbb{Q}$ in $\mathbb{R}$ and of $\mathbb{Q}(i)$ in $\mathbb{C}$ together with the continuity of the vector space operations.
Let's assume the complex case to save typing alternatives.
Since $\mathbb{Q}(i)$ is countable, $\operatorname{span}_{\mathbb{Q}(i)} S$ is countable if $S$ is countable.
Now let $M\subset E$ be a separable subset, and $D\subset M$ a countable dense subset. Then $F := \overline{\operatorname{span}_{\mathbb{Q}(i)} D}$ is the closure of a countable set, hence separable. Further,
$$\operatorname{span}_{\mathbb{C}} D \subset F$$
by the above, and since $\operatorname{span}_{\mathbb{Q}(i)} D \subset \operatorname{span}_{\mathbb{C}} D$, we have
$$F = \overline{\operatorname{span}_{\mathbb{C}} D},$$
so $F$ is in fact a closed linear subspace of $E$. Since trivially $M \subset \overline{D} \subset \overline{\operatorname{span}_{\mathbb{Q}(i)} D}$, $F$ is a subspace containing $M$, and hence $\operatorname{span}_{\mathbb{C}} M \subset F$. Thus $\operatorname{span}_{\mathbb{Q}(i)} D$ is a countable dense subset of $\operatorname{span}_{\mathbb{C}} M$.
