Convex Sets in Functional Analysis? 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!
 A: 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.

A: Mathematics is always a fine line between imposing conditions that are


*

*Strong enough to give you tools with which to show interesting results

*Weak enough to cover interesting applications

*Strange enough to not be easily analyzed or trivialized


The condition of local convexity happens to be


*

*Strong enough to leave you with a rich dual theory, due to the separation results. This in turn leaves you with interesting results such as the Krein-Milman theorem and a means of defining integration. While these are interesting results in their own right, they also turn out to be tools the are generally applicable to other problems.

*Weak enough to cover many applications to physics, PDE etc.

*Strange enough that many unexpected things happen.

A: 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 $\lambda\in\mathbb{R}$.
A: One key idea is the connection between balanced absorbing convex sets and (semi-)norms. They are basically one and the same.
For every norm the unit ball under that norm is a convex set. 
Similarly, any balanced bounded absorbing convex set generates a norm via the Minkowski functional (ie: how much do you have to scale the set to hit your point? Define that scaling factor as the norm of that point)
A: Besides the separation theorems, looking at this from the Krein-Milman and Choquet perspectives, the prettiness of the results alone justifies the importance of convexity, not to the mention their corollaries. Convexity requires a vector space structure...of course it is functional analysis.
As for semi-norms, they're immediate generalizations of norms. It's not that surprising that they characterize convexity.
A: To me, the primary relevance of convex spaces is in its very definition: When a space contains two points, it also contains the line segment connecting them. 
As for convex functions, one justification at calculus level is the uniqueness of local minimum. The most interesting theorem in "real analysis" regarding convex functions is the Jensen's inequality which statement in its probabilistic form should make it interesting enough.
