The finite rank-r projector domain is nonempty and compact

The diagonal matrix $P_{\ast }:=\mathrm{diag}(I_r,0)$ satisfies $P_{\ast }^{\ast }=P_{\ast }$, $P_{\ast }^2=P_{\ast }$, and $\mathrm{tr}{P}_{\ast }=r$, so it belongs to ${\mathcal{P}}_r$ by [F1]. It has rank $r$ by [F2], making the domain nonempty.

The equations $P=P^{\ast }$, $P^2=P$, and $\mathrm{tr}P=r$ are closed conditions on the real coordinates of ${\mathbb{H}}_n$, since adjoint, multiplication, and trace are continuous finite-dimensional maps. Thus ${\mathcal{P}}_r$ is closed. For $P\in {\mathcal{P}}_r$, $\mathrm{\parallel }P{\mathrm{\parallel }}_F^2=\mathrm{tr}(P^{\ast }P)=\mathrm{tr}(P^2)=r$, so it is bounded.

The real dimension of ${\mathbb{H}}_n$ is $n^2$: there are $n$ real diagonal entries and two real coordinates for each of the $n(n-1)\mathrm{/}2$ off-diagonal conjugate pairs. Therefore ${\mathbb{H}}_n$ is linearly homeomorphic to ${\mathbb{R}}^{n^2}$. By [F3], the closed bounded nonempty subset ${\mathcal{P}}_r$ is compact.