I have two somewhat related questions:
Question 1: What is the most general space (set of objects) on which we can do calculus?
Is it a normed space, or can we relax the conditions a bit further?
Subquestion 1a: Does it matter how we define "do calculus"?
I would think that the most basic idea of being able to do calculus would be being able to take limits. But does that necessarily imply that we can do something like differentiation and/ or integration?
Question 2: What is the least amount of structure we need to add to an affine space before we can do calculus?
A general affine space has no inner product or even a metric. Though one can compare straight line distances, I don't imagine that'd be enough to take limits with. But we could define some set of vectors in the associated vector space to be the "unit sphere" such that we'd be able to measure distances. Would that be enough, or would we need to add a inner product? In fact, maybe that wouldn't even be enough for practical purposes, maybe we'd need an inner product and for our space to be complete. Does anyone have an idea of the minimum amount of structure we'd need? Would there even be a way to show that less structure necessarily means we wouldn't be able to do calculus?