Solution
= Solution
Evaluation at zero is a <bounded linear functional> for the uniform <norm>. Define
$$
\boxed{\phi(f)=\sqrt{2\pi}\,(Tf)(0).}
$$
It is linear and satisfies $|\phi(f)|\leq\sqrt{2\pi}\|T\|\|f\|_\infty$. Since $e_n(0)=(2\pi)^{-1/2}$,
$$
\phi(e_n)=\sqrt{2\pi}\,\alpha_ne_n(0)=\alpha_n.
$$
The factor $\sqrt{2\pi}$ is required by the source's basis normalization.