If an Hilbert space is separable it exists a complete orthonormal sequence $\{e_k\}\subset\mathcal{H}$ and that means $$ x = \sum\limits_k \langle x, e_k\rangle e_k \,\forall x\in\mathcal{H} $$ so that the set of all $x$, meaning $\mathcal{H}$, should be a subset of $\text{span}(\{e_k\})$, defined as $$ \text{span}(\{e_k\}) \doteq \left\{ \sum\limits_k \alpha_k e_k :\alpha_k\in\text{C} \right\} $$ At the same time $\alpha_k e_k$ is an element of $\mathcal{H}$ for all $\alpha_k\in\text{C}$ and so it is every combination of these elements, meaning that $\text{span}(\{e_k\})\subseteq\mathcal{H}$.
By these inclusion relations I should conclude $\text{span}(\{e_k\})=\mathcal{H}$, but instead I see written $\text{cl}\,\text{span}(\{e_k\})=\mathcal{H}$ (see for example "Hilbert spaces with applications - Debnath, Mikusinki" third edition pag. 114, where span is defined with a finite summation, or the ending part of the proof of the theorem 4.8.15 pag. 185).
What is my incomprehension?
Often the linear algebraic span is defined to consist of all finite linear combinations of the form $\sum_{k=1}^n \alpha_k e_k$. In an infinite-dimensional Hilbert space, you may need an infinite series to express a vector; such a vector belongs to the closure of the span (i.e. it is a limit point). In a nutshell, working with the closure allows you to bring in concepts of convergence to deal with the possibility of being infinite-dimensional.