Computing the norm of $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$

Question

Let $p,q>1$, $\frac{1}{p}+\frac{1}{q} = 1$ and $(a_n)_{n \in \mathbb{N}}\in l_q$. Show that $\varphi : l_p \rightarrow \mathbb{R}$, $\varphi ((x_n)_{n \in \mathbb{N}})=\sum_{n=1}^{\infty} a_n x_n$ is a continuous linear functional and compute its norm.

My attempt:

By Holder's inequality:

$$|\varphi ((x_n)_{n \in \mathbb{N}})|= \bigg| \sum_{n=1}^{\infty} a_n x_n \bigg| \leq \sum_{n=1}^{\infty} |a_n x_n| \leq \|a\|_q \|x\|_p$$

Then $\varphi$ is continuous and $\| \varphi \| \leq \|a\|_q$. I'm having trouble to show that $\| \varphi \| = \|a\|_q$.

Any hints?

Answer 1

Consider $x_n= {\rm sign}(a_n)|a_n|^{q-1}$.