I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it!
Exercise: Let $1 \leq p,q \leq \infty$ be conjugate exponents. Let $a=(a_1,a_2,...)$ be a sequence such that $\sum_1^\infty a_n x_n$ converges for all $x=(x_1,x_2,...) \in l^p$. Prove that $a \in l^q$.
My idea is like this: It's sufficient to show that $a \in l^1$ since $l^1 \subset l^2 \subset... $ I define a family of operators $\{T_n \}_{n=1}^\infty$ by $T_n(x) = \sum_{k=1}^n a_k x_k$. It is clear that each $T_n$ is linear, bounded and that $\sup_{n}|T_n(x)| < \infty$ so by the Uniform Boundedness principle we get $\sup_n \| T_n \| < \infty$.
Is this a good approach? If so, I'd be grateful for some guidance on how to proceed to get to the conclusion:
$\sum_1^\infty |a_k| < \infty$
Otherwise, steer me in a better direction :)
Thanks in advance