Suppose that $X$ is a Banach space. Is there any way to explicitly compute the expression of norm of $X^*$ using the norm of $X$?
For example, consider the Banach space $X=\mathbb{R}^2$ equppied with the norm
$$\|(x,y)\|=\max\{|x|,|y|,|f(x,y)|\}$$
where $f(x,y):\mathbb{R}^2\to \mathbb{R}$ is a function of $x$ and $y$.
Now, what is the dual norm of the above norm, i.e., for an element $(a,b)\in X^*$, how to find $\|(a,b)\|_{X^*}$?