In functional analysis, what mean $$\overline{\text{span}(e_1,...,e_n)} \ \ ?$$
is it the closure of $\text{span}(e_1,...,e_n)$ ? How does it work ? Even if it's that, I don't understand how $\text{span}(e_1,...,e_n)$ can have a closure (because it's a vector space).
In functional analysis one of the main objects of study is the study of topological vector spaces. That is a vector space $V$ and a topology $\tau$ of $V$, such that the functions $$ V\times V \xrightarrow{+} V$$ and $$ \mathbb{C} \xrightarrow{\cdot v} V$$ are continuous. One of the main examples of topological vector spaces are normed spaces, where the topology is induced by a norm $\|\cdot\|:V \rightarrow V$. A sub examples of this kind are Banach spaces and Hilbert spaces.
$\overline{span(e_1,\ldots,e_n)}$ just means the closure of $span(e_1,\ldots,e_n)$ with respect to the topology of $V$.