Existence of an orthonormal basis on a closed subset of a Hilbert space

Question

In order to prove that if an orthogonal projection $\;P:\mathcal F \rightarrow \mathcal F\;$ is compact then $\;dimRan(P) \lt \infty\;$, our professor told us:

Since $\;P:\mathcal F \rightarrow \mathcal F\;$ is an orthogonal projection $\Rightarrow\;\;\;\;Ran(P)\;$ is a closed subset of $\;F\;$ (and so a Hilbert space). Thus, an orthonormal basis exists on $\; Ran(P)\;$.

My questions:

1. Why $\;Ran(P)\;$ is a Hilbert space? Don't we need $\;Ran(P)\;$ to be also finite dimensional in order to imply that?
2. Why does an orthonormal basis exist? Even if I assume $\;Ran(P)\;$ is Hilbert space, without the condition of separability, how do I get an orthonormal basis?

I 'm really confused. I would appreciate any help.

Thanks in advance!