Let $1<p$, $q<\infty$, $\frac{1}{p}+\frac{1}{q}=1$. Show that $(B^n_p)^* = B^n_q$.

Question

I am stuck on this question: Let $1<p$, $q<\infty$, $\frac{1}{p}+\frac{1}{q}=1$. Show that $(B^n_p)^* = B^n_q$. Should I apply the Hölder inequality here?

Answer 1

This will follow by Hahn-Banach Theorem \begin{align*} \|x\|_{X}=\max\{\left|\left<x,x^{\ast}\right>\right|: \|x^{\ast}\|_{X^{\ast}}\leq 1\}, \end{align*} and the Riesz Representation Theorem that $(L^{p})^{\ast}=L^{q}$ for conjugate $1<p,q<\infty$.