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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

Lang's definition of derivative

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.

Sadayuki's M-derivative

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?

Grassy LittleRoot
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  • I am by no means an expert on this, but I don't think there is any issue with extending the definition of differentiability to top. vector spaces in this way. But there are some meaningful results that may only hold under stronger conditions. For example, the standard proof of showing that a function with zero derivative is constant is to use the Hahn-Banach theorem to reduce to the real-valued case, which only works for locally convex spaces. – Christopher A. Wong Apr 17 '17 at 05:09
  • Hm... I see. I think Sadayuki shows the mean value theorem for the $M$-derivative on locally convex spaces. I understand that giving more structure makes theorems appear and puts more intuition on the thing, but I guess the content of my question is more of a philosophical thing. I want to understand what is the more intrinsic thing about this differentiability. What are the difficulties behind my idea. – Grassy LittleRoot Apr 17 '17 at 09:47
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    I guess what I'm trying to say is that I just want to see some crazy stuff. How can we push this? – Grassy LittleRoot Apr 17 '17 at 09:57

1 Answers1

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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.

  • From what I understood from the article, they do this differential calculus on metric spaces seeing the metric space as a manifold. Just to define manifolds (in the usual way with atlas, charts etc) you need to talk about differential calculus in the spaces that the manifold is going to be modeled. In the case you mentioned, it's a manifold modeld on $\mathbb{R}^{n}$. Lang models his manifolds on banach spaces. From what I saw you can change it to topological vector spaces and the definition still seems valid. – Grassy LittleRoot Apr 20 '17 at 01:51
  • I want to know if it can go futher in the direction of taking away more properties of the topological vector space. I don't know if going in the direction of locally ringed spaces is the proper way to do this, or if this can still be done using an maximal atlas as differentiable structure. – Grassy LittleRoot Apr 20 '17 at 01:55