Do the dual spaces satisfy equivalence relation? For example the dual space of $\mathbb{R^n}$ is $\mathbb{R^n}$. Do the remaining two properties of equivalence relation can be applied to dual spaces, in general. that is,
$Reflexivity, Symmetry, Transitivity$.
Note: The dual space of $l^1$ is $l^\infty$. Can the dual space of $l^\infty$ be $l^1$?
I don't have any clue about the generalization of these properties. Any efforts will be appreciated, thanks in advance.