Projections of functions and bases

[member 428835]

Homework Statement

On $L_2[0,2\pi]$ where $e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}$. Given $f(x) = x$, find $Pr_e f$.

Homework Equations

See solution.

The Attempt at a Solution

I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?

 

[Dr Transport]

That's how I'd approach it.