Determine the polinomial function p(t) belonging to $\mathbb{R}_2[t]$ so that the value of
$\int_{0}^{2\pi} [p(t)- \cos(t)]^2 dt$
is minimal.
So as this is problem of linear algebra my first thought was to find the projection of the integral in $\mathbb{R}_2[t]$ .
However I need an orthogonal basis for $\mathbb{R}_2[t]$...
But when I tried to do it using Gram-Schmidt process I reach to the vectors $(1,0, -t^2$) and 1 and $-t^2$ are not orthogonal so I must be applying it wrongly...
Can someone plese help finding out this... The fact that we have a integral instead of vectors is making me confused.
Thank you!