What is Convex about Locally Convex Spaces? This might be a silly question, but what motivates the name "locally convex" for locally convex spaces?
The definition in terms of semi-norms seems to have nothing to do with convexity or with the other definition involving neighborhood bases -- and the neighborhood basis definition makes little sense to me either, because it refers to sets which are "absorbent", "balanced", and convex. 
Why the restrictions? And then why aren't they called "locally absorbent, balanced, and convex spaces"? And why do we never here about the terms absorbent or balanced in any other context?
Also, I know that Banach spaces are locally convex, but this just confuses me further -- what do Banach spaces have to do with convexity? And why are locally convex spaces a natural generalization of Banach spaces?
I have some vague ideas -- the Hahn-Banach theorem (and hyperplanes) are used a lot in convex programming, and the "p-norm" is only a norm for $p \ge 1$, the same values for which $x^p$ is a convex function -- are norms somehow "convex", does this follow from the triangle inequality? Then why aren't arbitrary complete metric spaces locally convex?
Any insights would be greatly appreciated.
 A: The only reason the question seems silly is that you include the answer! A locally convex TVS is one that has a basis at the origin consisting of balanced absorbing convex sets. The reason for the emphasis on "convex" is that that's what distinguishes locally convex TVSs from other TVSs: every TVS has a local base consisting of balanced absorbing sets.
Regarding "Also, I know that Banach spaces are locally convex, but this just confuses me further -- what do Banach spaces have to do with convexity? And why are locally convex spaces a natural generalization of Banach spaces?", surely this is clear. Balls in a normed space are convex (as well as balanced and absorbing).
Why aren't arbitrary complete metric spaces locally convex? Huh? The notion of convexity makes no sense in a general metric space.
A: There is two different yet equivalent definition of Locally convex spaces : one in which the topology endowed by a family of semi-norms, and one in term of absorbent balanced and convex basis. The equivalence between the two definition is rather long to prove but you can find it in Rudin's Functional Analysis. 
This might be of interest to you : https://en.wikipedia.org/wiki/Minkowski_functional
Why are locally convex spaces a natural generalisation of Banach spaces ? The topology of a Banach space is induced by the norm. And a norm is a family of semi-norm, namely the family containing one semi-norm which happens to be a norm.
What do Banach spaces hae to do with convexity ? Every $x$ in a Banach space has a localy base of convex sets, namely the balls centered a $x$ and of radius $r > 0$. Every open neighbourhood of $x$ contains one such ball centered at $x$.
