I am trying to understand a problem in folland's real analysis.
In a norm linear space.
There is no slowest rate of decay of the terms of an absolutely convergent series, that is, there is no sequence $\{a_n\}$ of positive number such that $\sum a_n |c_n| < \infty$ iff $\{c_n\}$ is bounded.
I am confused by the problem, take $\mathbb{R}$ for example, I can certainly let $\{c_n\}$ be $\{\frac{1}{n^2} \}$, and $\{a_n\}$ be $\{1\}$, then $\sum a_n |c_n| < \infty$ and $\{c_n\}$ is bounded and $\{a_n\}$ are positive, I am certainly misinterpreting the problem, but I'm not sure what I'm misinterpreting, any help with clarifying my confusion would be helpful, thanks!
Your sequence $a_n=1/n^2$ does not work: $\sum_{n}a_n|c_n|$ is convergent even for $c_n=\sqrt{n}$ which is not bounded.
P.S. This is an interesting reference.