  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.
