Suppose that $\varphi$ is an element of the dual of $\ell_2$. Then I need to show that the sequence $$(\varphi(e_1),\varphi(e_2), \varphi(e_3), \cdots)$$ is in $\ell_2$ (where $\{e_k\}$ is of course the standard basis for $\ell_2$).
So far, I've found that $$ \begin{align*} \sum_{k = 1}^N |\varphi(e_k)|^2 &= \sum_{k = 1}^N \varphi(e_k) \varphi^*(e_k) \\ &= \varphi\left(\sum_{k = 1}^N e_k\varphi^*(e_k) \right) \end{align*} $$ This seems promising, but the best I can seem to do in order to bound the right-hand side is $$ \begin{align*} \varphi\left(\sum_{k = 1}^N e_k\varphi^*(e_k) \right) &\leq \Vert \varphi \Vert \cdot \left\Vert \sum_{k = 1}^N e_k \varphi^*(e_k) \right\Vert \\ &\leq \Vert \varphi \Vert \cdot \left( \sum_{k = 1}^N \varphi^*(e_k) \right)^{1/2} \end{align*} $$ which appears to go $\infty$ as $N \to \infty$. How can I find a better bound?