Notation query: matrix projection

Question

I've used the following notation in a report:

$G\boldsymbol{w}\in\mathcal{H}_2\,\forall\boldsymbol{w}\in\mathcal{H}_2 $

In other words $G$ projects $\boldsymbol{w}$ back into the same set. But this notation doesn't feel right. Is this correct or could somebody suggest better / clearer / more appropriate notation?

Answer 1

I collect three common options, ranked by my own preference.

1. $G(\mathcal H_2)\subseteq \mathcal H_2$
2. $G(w) \in \mathcal H_2$ whenever $w\in \mathcal H_2$
3. $G:\mathcal H_2\to \mathcal H_2$