Here is a question I am hoping a friendly person can clarify for me:
Consider the inner product space V with inner product $\langle f,g \rangle = \frac1\pi\int_{-\pi}^\pi f(x)g(x)dx.$ Let $B=[\frac{1}{\sqrt2},cos(x),cos(2x),...,cos(nx),sin(x),sin(2x),...,sin(nx)]$.
I proved that B is an orthonormal set.
I am wondering: is the dimension of the subspace $W=span(B)$ equal to n+1? I figured this might be correct because of the relationship between sinusoids.
Secondly, for the case n=1, how would I find the orthogonal projection of $f(x)=x$ in W?
Thank you for your assistance.