Let $A$ be a directed set, and $\ell^\infty_A$ the (complex vector) space of all
eventually bounded nets $A\to \mathbb{C}$. We can define the *limit superior* seminorm on $\ell^\infty_A$:
$$
\vert\vert{(u_\alpha)}\vert\vert:= \inf_{\alpha\in A} \sup_{\beta\geq \alpha} |u_\beta|\ .
$$
This is indeed finite, and we can check (unless I am mistaken) the inequality
$\vert\vert{u+v}\vert\vert\leq \vert\vert{u}\vert\vert+\vert\vert v\vert\vert$ by an $\epsilon/2$ argument.

Now my questions are :

1) Is $\ell^\infty_A$ a complete seminormed space ?

2) Is the subspace $c_A$ of convergent nets $A\to \mathbb{C}$ a closed subset of $\ell^\infty_A ?$.

3) Is there a reference for these type of questions ? Is the space $\ell^\infty_A$ (for general directed set $A$) used or studied somewhere in the litterature?