Reference request for Fenchel-Rockafellar duality for dual system In the Fenchel-Rockafellar duality theory, the usual setup contains: $X$-Banach space, $X^*$-continuous dual space of $X$, $\langle x^*, x\rangle := x^*(x)$ where $x^*\in X^*$ and $x\in X$.
On the other hand, I also know that there is a theory called dual system, where we may establish the dual structures using: $X$ a Banach space, $Y$ another Banach space, $\langle x, y\rangle=a(x, y)$ where $a: X\times Y \longrightarrow \mathbb R$ is a  bilinear map.
So, you may see a correspondence.




Classical
Dual system




$X$
$X$


$X^*$
$Y$


$\langle x^*, x \rangle := x^*(x)$
$\langle x, y \rangle$ bilinear




The advantage of using dual system is on the symmetric role of $X$ and $Y$. So $X$ and $Y$ are interchangable. This is not true for $X$ and $X^*$ since $(X^*)^*$ may differ from $X$ (in this case $X$ is said to be non-reflexive). I therefore think that, the dual system should be considered as a better/more general setup for Fenchel-Rockafellar duality theory.
Could anybody provide a book, which builds the Fenchel-Rockafellar duality on top of dual system (e.g. a dual system with symmetric role of $X$ and $X^*$)? Thanks!
 A: Actually for a dual system it suffices $X$ and $Y$ to be linear spaces, i.e., no apriori topological structure.They get their topological structures from the weak topologies.
In the presence of (Hausdorff) separation $Y$ can be identified with the topological dual of $X$ so a template for all dual systems is provided by the classical $(X,X^*)$ where $X$ is a locally convex space and $X^*$ is its topological dual.That is why it is enough to build the convex conjugate of a function using only the template.
In my book the Fenchel-Rockafelar duality refers to functions.
There are several books on the subject of dual systems e.g.
Bourbaki, Espaces Vectoriels Topologiques
Edwards, Functional Analysis: Theory and Applications
If you are interested in the convex conjugate then you can pick any convex analysis book starting with the book of Rockafellar for finite dimensional spaces and ending with Borwein or Zalinescu.
P.S. If you want the symmetry between $X$ and $Y$ you lose the topology, from strong to weak.
