# Duality between Banach spaces and compact convex spaces

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.

• By "convex compact space" you mean "convex compact subset of a locally convex space"? – Dirk Oct 24 '17 at 13:52
• @Dirk : Not exactly, I'm hoping for an 'intrinsic' definition, i.e. something independent of any embeddings into a vector space, even if the duality would automatically implies the existence of such an embeddings. Like a compact topological space with some operations on it (typically an algebra for a Giri monad) possibly satisfying some additional condition like cancelativity. The notion mentioned in Robert Furber answer seem to be what I have in mind, but I havn't look the paper he mentions yet. – Simon Henry Oct 24 '17 at 13:57
• Then what is a linear functional on a convex compact space? – Dirk Oct 24 '17 at 14:20
• @Dirk : A linear function of norm smaller than $K$ would be a morphism of algebras to the interval $[-K,K]$ which should have itself a natural algebra structure for the chosen monad. I.e. it will be a real valued continuous function on the compact space which is compatible to the algebraic structure required (this algebraic structure being suppose to correspond to the choice of a zero element and some notion of 'barycenter') – Simon Henry Oct 24 '17 at 14:24
• @Dirk: More elementary: One can define a convex space abstractly by formalising the notion of a (finite) convex combination. A affine map is then one which preserves convex combinations. If you have a pointed compact space and call the special point 0, then a linear map is an affine map that maps 0 to 0. See ncatlab.org/nlab/show/convex+space for example (which of course is even more general than what I explained) – Johannes Hahn Oct 24 '17 at 19:13

## 3 Answers

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.

Judging by what you say in the question, I think you are referring to what Świrszcz called "Saks spaces" in this article:

Monadic functors and convexity, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 22 (1974), 39–42

There they are defined to be pointed compact convex sets (embeddable in a locally convex space), subject to the condition that for every point $x$ there exists a $y$ such that $\frac{1}{2}x + \frac{1}{2}y$ is equal to the base point (this does not match what everyone else uses "Saks spaces" to mean). Świrszcz proved that this category is equivalent to the category of Eilenberg-Moore algebras of the monad of signed (Radon) measures with total variation $\leq 1$. Incidentally, Świrszcz's article predates Giry's article, so it is perhaps not quite right to refer to it as a Giry monad.

A more extensive version appeared as a preprint. I have a copy, but it is hard to track down by the official channels.

However, Świrszcz does not touch on the duality with Banach spaces. This is covered by the reference given by Seine for Waelbroeck spaces. It is not hard to show that "Saks spaces" in Świrszcz's sense are equivalent to Waelbroeck spaces by taking a "Saks space" $X$, embedding it in a locally convex space $E$, then shifting the base point of $X$ to 0 and restricting $E$ to the subspace absorbed by $X$. There are also other versions of this duality, such as Smith spaces (maybe Sergei Akbarov can give a reference).

Yes, see Proposition 8.1.3 of the book Mathematical Quantization. This describes a duality between Banach spaces and "dual unit balls", which are defined as compact, convex, balanced subsets of locally convex TVSs.