Let $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n\in\{2,3\}$, and let $$H(\operatorname{div};\Omega):=\{v\in L^2(\Omega):\operatorname{div}v \in L^2(\Omega)\}.$$
My question is: does $H(\operatorname{div};\Omega)$ have a Schauder basis?
If the answer is "yes", I can justify the existence of a collection of projectors $P_m:H\to H_m$ uniformly bounded that converges pointwise to identity, where $\{H_m\} $ is a increasing sequence of finite dimentional subspaces of $H(\operatorname{div};\Omega)$ ( this is possible 'cause this greater space is separable).
$H(\operatorname{div};\Omega)$ is Hilbert (I remember answering this question not long ago).
Every Hilbert space has an orthonormal basis (you must have known this fact). The orthonormality (orthogonality+normalized) is with respect to the $H(\operatorname{div})$-inner product: $$ (u,v)_{L^2 } + (\operatorname{div} u,\operatorname{div}v)_{L^2 }. $$
Prove this basis is the Schauder basis (straightforward using inner product).
To answer your second question: yes, you can find this projection sequence, converges to identity, in the sense of $H(\operatorname{div})$-norm: $$ \|P_m v - v\|_{H(\operatorname{div})} \to 0. $$ This is the ground for finite element method.