A simple observation about compactness and fast decay of Fourier coefficients

Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that $|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every $y\in Y$. We prove this result and we use it for the study of fast decay of Fourier coefficients in $L^p(\mathbb{T})$ and frame coefficients in the Hilbert setting.