Let $W$ be a finite-dimensional subspace of an inner product space $V$, and let $E$ be the orthogonal projection of $V$ on $W$. Prove that $(E(\alpha),\beta) = (\alpha,E(\beta))$ for all $\alpha,\beta\in V.$
Its exercise from section $8.2$ so we just have to use only definition of inner product space and properties of orthogonal projection.
I am trying it as $(E(\alpha),\beta)=(\alpha-\alpha+E(\alpha),E(\beta)-E(\beta)+\beta)=(\alpha,E(\beta))-(\alpha,E(\beta))+(\alpha,\beta)-(\alpha,E(\beta))+(\alpha,E(\beta))-(\alpha,\beta)+(E(\alpha),E(\beta))-(E(\alpha),E(\beta))+(E(\alpha),\beta)$ but this is not useful. Please suggest. Thanks.
Hint. Decompose $\alpha$ into the sum of $E(\alpha)\in W$ and $\alpha'=\alpha-E(\alpha)\in W^\perp$ and similarly decompose $\beta$ into the sum of $E(\beta)\in W$ and $\beta'=\beta-E(\beta)\in W^\perp$. Note that $(u,v)=0$ if one of $u,v$ belongs to $W$ and the other belongs to $W^\perp$.