Consider a sequence of functionals $(f_n)$, $f_n(x)=\int_{-1}^1x(t)\cos(nt)dt,\ n\geq 1$, on the space $L_2(-1,1)$. I need to prove that $f_n(x)\to 0$, as $n\to\infty$, for all $x\in L_2(-1,1)$.
I know that $\int_{-1}^1\cos(nt)dt\to 0$, as $n\to\infty$, and I tried to extract this term as a multiplicand (applying Holder's inequality) but with no success.
Or maybe I should take into account that $L_2(-1,1)$ is a Hilbert space and somehow use the general form of continuous linear functionals there?