Every orthogonal projection is a Hermitian operator?

Question

I am having a hard time with this proof

Prove that every orthogonal projection is a Hermitian operator. That is, if $V$ is an inner product space with subspace $W,$ prove that $T(v) = \operatorname{proj}_w (v)$ is a Hermitian operator.

Answer 1

Let $f_{1}=g_{1}+h_{1}$, $f_{2}=g_{2}+h_{2}$ for $g_{1},g_{2}\in W$ and $h_{1},h_{2}\in W^{\perp}$, then \begin{align*} \left<Pf_{1},f_{2}\right>&=\left<g_{1},g_{2}+h_{2}\right>\\ &=\left<g_{1},g_{2}\right>\\ &=\left<g_{1}+h_{1},g_{2}\right>\\ &=\left<f_{1},Pf_{2}\right>. \end{align*}