What is the absolute minimum needed on a space to do differential calculus?

Question

A long time ago, I came across Lang's Differential Manifolds. Besides his definition of manifold, one thing that made me pretty nuts was this

So after studing some real analysis and more linear algebra and topology, I came back to this page and began questioning this definition. He assumes all topological vector spaces in the text are going to be what he calls banachable (can be given a norm that induces the given topology and is complete in such norm). So I thought this definition was only a reestatement of the definition using the norm, but without the norm. But I really didn't understand this definition, this concept of tangency.

After some more research I found Sadayuki Yamamuro's Differential Calculus in Topological Vector Spaces.

Not only I can understand this concept (somehow) but I could translate this definition to topological vector spaces over any topological field. This concept of $M$-differentiation generalizes the Fréchet and (linear) Gâteaux derivatives and much more.

So my question is: how far can this concepts of differentiability be generalized? Is Lang's definition with tangency to $0$ valid not only for banachable ones, but for arbitrary topological vector spaces? Is the $M$-differentiation the farthest we can go? Can something along this lines be done with modules?

Answer 1

One can generalize differential calculus in multiple directions: for example,

• to vector spaces without a metric structure
• to metric spaces without a vector structure

Such extensions are obviously not comparable: the set of generalizations is not totally ordered. A standard references for the latter is Nonsmooth calculus by Heinonen.