We have $L: \ell^{2} \to \mathbb{K},$ where $\mathbb{K}$ denotes real or complex space. And $L(x) = \sum_{n=1}^{\infty}\frac{x_{n}}{n^{q}}.$ What I've tried so far is the following.
$ \|L(x)\| = \| \sum_{n=1}^{\infty}\frac{x_{n}}{n^{}} \| \leq \sum_{n=1}^{\infty} \| \frac{x_{n}}{n^{}} \| = \sum_{n=1}^{\infty} \lvert \frac{1}{n^{}} \rvert \lvert x_{n}\rvert.$ But I'm not sure how to proceed from here. Any help is appreciated.
I think you are nearly there and just need to use the Cauchy Schwarz inequality. In particular, $$||L(x)||\leq \sum_{n=1}^\infty \left|\frac{1}{n}\right|\, |x_n|\leq ||x||_{l^2} \left(\sum_{n=1}^\infty \left|\frac{1}{n}\right|^2\right)^{1/2}=||x||_{l^2}\left(\sum_{n=1}^\infty\frac{1}{n^2}\right)^{1/2}<\infty.$$ Since $x\in l^2$ and $\sum_{n=1}^\infty 1/n^2$ converges by the $p$ test!