 Solution

For each fixed $n$, the coefficient functional is bounded sinceSuppose $f_j\to f$ and $T{f}_j\to g$ uniformly. Taking coefficients and passing to the limit givesfor every fixed $n$. No uniform bound on ${\alpha }_n$ is needed for this passage. Uniqueness gives $g=Tf$, so $T$ has a closed graph. The space $X$ is Banach, being the closed subspace defined by $f(-\pi )=f(\pi )$ inside the continuous functions with the uniform norm. The closed graph theorem therefore proves $T$ is bounded.