[Math] Convex Sets in Functional Analysis

Question

Why did Bourbaki choose to study convex sets, convex functions and locally convex sets as part of the theory of topological vector spaces, and what is so important about these concepts?

I'd like to really feel the intuitive reason why they devoted an entire chapter to these things, to appreciate the necessity for studying them here and not somewhere else, why they are naturally related to semi-norms and weak topologies, and why lead to something so important as the Hahn-Banach theorem.

(Contents of the chapter viewable on amazon if necessary)

Edit – to be clear: I'm not interested in ex post facto justifications for studying convexity. You could make the same arguments about e.g. point set topology, missing the fundamental simplicity in the fact that topology is just about 'near-ness', ignoring how every single concept/theorem has a deep intuitive interpretation as such. I'm interested in the most core fundamental conception of convexity as it lies within the edifice of mathematics as a whole, in the sense that one would be able to derive the contents of the chapter themselves when viewed from the right perspective.

Thanks!

Answer 1

Maybe the following can at least partly answer your question. First we look at

topological vector spaces in increasing generality:

\begin{align*} &\text{finite dimensional vector spaces - Hilbert Spaces - Banach Spaces}\\ &\text{Frechet Spaces - locally convex vector spaces}\\ \end{align*}

as Jänich does in his Topology. There he mentions an example from Dieudonnes Treatise on Analysis Volume II of a locally convex, but not metrizable and so not pre-Frechet topologial Vector Space.

He states, that these non-metrizable spaces occur naturally in functional analysis for example when we want to find on a given topological Vector Space $E$ the weak topology, i.e. the coarsest topology, so that all continous, linear mappings are continous. He further states, that if $E$ is an infinite dimensional Hilbert Space, then $E$ equipped with the weak topology is already a locally convex Hausdorff Space, but not metrizable.

And now some deeper information about this natural wish to develop a theory around locally convex vector spaces.

Here's an extract from Bourbakis Elements of the History of Mathematics. He writes in the end of chapter 21: Topological Vector Spaces:

Excerpt from Bourbakis Elements of the History of Mathematics ch. 21:

It had on the other hand been observed, before 1930, that notions such as simple convergence, convergence in measure for measurable functions, or compact convergence for entire functions, are not capable of being defined by means of a norm; and in 1926, Fréchet had noted, that vector spaces of this nature can be metrizable and complete.

But the theory of these more general spaces was only to develop in a fruitful way in combination with the idea of convexity. This latter (that we saw appearing with Helly) was an object of study for Banach and his pupils, who recognised the possibility of interpreting thus in a more geometric way numerous statements of the theory of normed spaces, preparing the way for the general definition of locally convex spaces, given by J. von Neumann in 1935. $\ldots$

Finally and especially, it is certain that the main impulsion that motivated this research came from new possibilities for applications to Analysis, in domains where Banach Theory was inoperative: the theory of sequence spaces must be mentioned in this context, developed by Köthe, Toeplitz and their pupils since 1934 in series of memoirs, the recent setting up of the theory of analytical functionals of Fantappié, and above all the theory of distributions of L. Schwartz, where the modern theory of locally convex spaces found a field of applications that is without doubt a long way from being exhausted.