What are the implications of the dual norm of a norm?

Question

I was doing a series of questions proving that the dual norm of $l_p$ is $l_q$, where $p,q$ satisfies $\frac{1}{p} + \frac{1}{q} = 1$.

I was able to prove this result but I do not see the point of dual norm.

What is the implication, for example, that the dual norm of $\|\cdot\|_2$ is itself? Or that the dual norm of $\|\cdot\|_1$ is $\|\cdot\|_\infty$?

How can we use this result in some way?

Answer 1

The major implication is that you know the space of continuous linear functionals on the space (for $1<p<+\infty$), and you can, therefore, apply all the properties of $\ell_{q}$. Among other things this allows you to introduce weak topology, which, in turn, sometimes is the natural topology for the space of solutions of an equation that you study.

The particular case $p=2$ gives you another view on the Riesz representation theorem - because we know that $\ell_2$ is a Hilbert space.