How to show this sequence is in $l^p$

Question

Suppose $\{a_k\}$ is a sequence such that for any sequence $\{b_k\}$in $l^q$ series $\sum_k a_k b_k$ is convergent, then how to show $a_k$ is in $l^p$ where $\infty>p,q>1$ and $1/p+1/q=1$.

Answer 1

Define $T_n:l^{q}\to \mathbb C$ by $T_n \{b_k\}=\sum_{k=1}^{n} a_k b_k$. If we show that $||T_n||=(\sum_{k=1}^{n}|a_k|^{p})^{1/p}$ we can finish the proof using Uniform boundedness principle. From now on I will take all sums from $1$ to $n$. It is easy to see that $||T_n||\leq (\sum_{k=1}^{n}|a_k|^{p})^{1/p}$ . Let $b_k=\frac {|a_k|^{p}} {a_k}$ if $a_k \neq 0$ and $0$ otherwise. Note that $\sum |b_k|^{q}=\sum |a_k|^{(p-1)q}=\sum |a_k|^{p}$. Hence $||T_n|| \geq \frac {|T_n \{b_k\}|} {(\sum |a_k|^{p})^{1/q}}$. But $T_n \{b_k\}=\sum |a_k|^{p}$. Can you take it from here?