Notation query: matrix projection

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?

• $G:H_2\to H_2$? – TZakrevskiy Mar 19 '14 at 15:04
• @TZakrevskiy is that not then a definition, rather than a single rule? $G$ is not limited to the elements of $\mathcal{H}_2$. – oLas Mar 19 '14 at 16:58
• I think that as long as the statement of this property is clear and concise, it is ok. Another option would be to write $G(\mathcal H_2)\subset \mathcal H_2$ or say "$G$ is an endomorphism on $\mathcal H_2$", or "$G\in\mathcal L (\mathcal H_2)$", or something else. Though personally I prefer the way you wrote it. – TZakrevskiy Mar 19 '14 at 17:33

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