If X is the space of ordered n-tuples of real numbers and $\displaystyle ||x||=max|{\xi}_{j}|$ where $\displaystyle x = ({\xi}_{1 }.......,{\xi}_{n}) $, what is the corresponding norm on the dual space X'?
$\displaystyle x'\in X'$ can be represented by $\displaystyle (a_1,\ldots,a_n)\in\mathbb{R}^n$ (we have $\displaystyle \langle x',x\rangle =\sum_{j=1}^na_jx_j$). We have $\displaystyle \lVert x'\rVert=\sup_{\lVert x'\rVert =1} \left|\langle x',x\rangle\right|$ and taking $\displaystyle x_j = \mathrm{sgn}a_j$ we find $\displaystyle \lVert x'\rVert =\sum_{j=1}^n|a_j|$.