Find projection operator $P_{E}:H\rightarrow E \subset H$, H - Hilbert space

Question

In Hilbert space $H=L_{2}(-\pi,\pi)$ we have three dimensional subspace $E:=\textrm{span}_{R}\{t,t^{2},sin(t):t\in[-\pi,\pi]\}$.

I have to find operator $P_{E}:H\rightarrow E \subset H$, its norm $||P_{E}||$ and distance $d(\sin(2t),E)$, note that $\sin(2t)\in H$. I would like to ask for tips on how to start a task.

Answer 1

Hints: Use Gram -Schmidt Process to find an orthonormal set $\{f,g,h\}$ which spans $E$. This is the main step. Everything else will then follow from general theory: $P_E(F)=\langle F ,f \rangle f+\langle F ,g \rangle g+ \langle F ,h \rangle h$. $\|P_E\|=1$ (since any non-zero projection has norm $1$). The distance of any $F$ from $E$ is nothing but $\|F-P_E(F)\|$.