A proof pertaining to the projector operator Let $H_{1}$ be any subspace of a Hilbert space $H$, and let $H_{2} = H_{1}^{\bot}$ be the orthogonal complement of $H_{1}$, so that an arbitrary element $h \in H$ has a unique representation of the form $h = h_{1} + h_{2}$ where $h_{1} \in H_{1}$ and $h_{2} \in H_{2}$. Let $Ph = h_{1}$. $P$ is a continuous linear operator known as the projection operator. I want to prove that the projection operator is completely continuous if and only if the subspace $H_{1}$ is finite dimensional. 
I have been thinking about this one for awhile now and have no idea. I would really appreciate the help!!! 

Suppose the range of $P$ is infinite dimensional. Then there exists an orthonormal sequence $\{ e_{n} \}_{n=1}^{\infty}$ in the range of $P$. This is a sequence of unit vectors for which $\{ Pe_{n}=e_{n} \}_{n=1}^{\infty}$ has no convergent subsequence.
