Let $g_{1} = \frac{1}{\sqrt{2\pi}}$, $g_{j}=\frac{sin(jt)}{\sqrt{\pi}}$ and $h_{j}=\frac{cos(jt)}{\sqrt{\pi}}$ $s_{n}(g(t)) = <g_{1},g >g_{1} + \sum_{j=1}^{n} <g_{j},g >g_{j} +\sum_{j=1}^{n} <h_{j},g >h_j$
where the union of $g_{1}, g_{j}, h_{j}$ forms an orthonormal sequence in $L^{2}(\pi,\pi)$.
QUESTION:
If $s_{n}:L^{2}(\pi,\pi) \to L^{2}(\pi,\pi)$, then how can I prove $s_{n}$ is in fact in $L^{2}(\pi,\pi)$?
My Attempt:
Since I have to check that $\int_{\pi}^{\pi} |s_{n}g|^{2} < \infty$ I tried to calculate $<s_{n}g,s_{n}g>$ to be able to apply Bessel inequality somewhere but I didn't do much until I had problems with the eliminations of the terms.
Any help would be enormously appreciated. <3
$s_n$ is a linear combination of the functions $\sin (kt)$ and $\cos (kt)$ and each of these are in $L^{2}$. The inner products in the definition of $s_n$ are just scalars and you don't need any information about them.