If I have $ A = \{a \in \ell_2 : |a(n)| \leqslant c(n)\}$ for $c(n)\geqslant 0$ where $ n \in N $, and I want to show that is $A$ compact in $\ell_2$ iff $\sum{c(n)^2}<\infty$. How do I go about showing both directions?
If $f \in C(T)$ is the $1$-periodic continuous functions in $\Bbb R$, how to show $\lim \limits_{|n|\to\infty}\int_0^1 e^{-2\pi inx}f(x)=0 ?$ Also is this true if $f$ were in the closure of the set of 1-periodic step functions in $R$ Intuitively, I think the later is false since boundedness does not imply continuity.