Linear continuous form $\phi$, pre-Hilbert space $E$

Question

Let $E$ a pre-Hilbert space of the complex sequences $(u_n)$ which satisfy $\exists N \in \mathbb{N}, \forall n \geq N,\ u_n=0$ with the inner product $(u,v) = \sum_{n=0}^{\infty} u_n \overline{v_n}$.

I have to show that $\phi : E \rightarrow \mathbb{C}$ defined by $\phi(u) = \sum_{n=1}^{\infty} \dfrac{u_n}{n}$ is a linear continuous form on $E$.

To do it, I have to use Cauchy-Schwarz... But I don't know how to use it for this question. Someone could help me ?

Answer 1

It's straightforward: you have $$|\phi(u)|\leq\sum_n|u_n|\,\frac1n\leq\left(\sum_n|u_n|^2\right)^{1/2}\,\left(\sum_n\frac1{n^2}\right)^{1/2}=\frac{\pi}{\sqrt6}\,\|u\|.$$

Answer 2

The sequence $(u_{n})$ is compactly supported, so $(u_{n})\in l^{2}$, and hence one uses Holder's inequality to conclude that $|\phi(u)|\leq\|(u_{n})\|_{l^{2}}\|(1/n)\|_{l^{2}}<\infty$.