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!
In my experience, convex sets are very important (at least in functional analysis and optimization) because of the various separation theorems that apply to them.
For example, if $A$ and $B$ are convex, closed subsets of a Banach space, and $A$ is compact, then there exist a linear functional $\ell$ such that $\ell(A)\le\lambda<\ell(B)$ for some $\lambda\in\mathbb{R}$.