We have recently proven in class that $\ell_q \cong \ell_p^*$, and I would like to understand why this is valuable. I know that isomorphism (and general structure preserving mappings such as homeomorphism etc.) are useful for performing operations in a given space by moving to an 'easier' space, and then returning to the original one as though one has performed the operations on it in the first place. However, in Group Theory, for example, it is easy to see this is useful, but how does it transpire in $\ell_p$? Are there any simple/common examples that demonstrate why it is beneficial to have the dual?
Thanks.
What is arguably even more important is that functionals in $\ell_p^*$ have the same norm as their $\ell^q$-representative. With that you can compute the norm of functionals like $T:\ell^2 \rightarrow \mathbb{R}$ $$ Tx := \sum_{k = 1}^{15} x_k. $$ The representative in $\ell^2$ is $$ \sum_{k = 1}^{15} e_k $$ and has norm $\sqrt{15}$. So the norm of $T$ is $\sqrt{15}$. Since the analogue also holds for $L^p$ you can also prove nice theorems like $$ f \in L^p(\Omega) \iff f\in L^1_{loc}(\Omega) \text{ and }\exists C>0 \text{ s.t. } \left \lvert \int_\Omega \xi f ~\mathrm{d}x \right \rvert \leq C \lVert \xi \rVert_{L^q(\Omega)}\quad \forall \xi \in C_0^\infty(\Omega) $$ that characterize $L^p$.
I would suggest that the spaces $\ell^p$ are not of that much value in functional analysis. The rather serve as nice toys to gain intuition on some concept because they are easy but not finite-dimensional.