I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras for a certain Giri monad acting on the category of compact topological spaces, maybe with additional conditions) given by:


To a Banach space one associate the unit ball of its dual in the weak\* topology. And to a convex compact space one associate the space of linear function on it, with the uniform norm.


I can't remember if this is something I read somewhere or just something I imagined due to other weaker results in this direction (like the fact that every banach spaces is isomorphic to the space of Weak\* continuous linear form on its dual).


So, do we indeed have a duality of this kind ? Has this been worked out somewhere, or is it some sort of folk results ? does it have a name ?


PS: I had too chose one answer to accept, but all three answers were all equally good and very interesting. 



Duality between Banach spaces and compact convex spaces
In my opinion, the most natural expression of this symmetric duality is that between Banach spaces and Waelbroeck spaces (the latter being Banach spaces with a suitable compact topology on the unit ball). This goes back to work of Lucien Waelbroeck and Henri Buchwalter. For a systematic treatment, I would refer to the monograph on category theory and Banach spaces by Cigler, Losert and Michor which can be readily found online.


