Calculus in Vector Spaces without Norm

As emphasized by J. calculus primarily deals with the approximation (in a neighborhood of some point) of given mappings of vector spaces by linear mappings. The approximating linear map has to be a "good" approximation in some precise sense: it has to be "tangent" to the given map. A very useful notion of "tangent" can easily be introduced for maps between normed vector spaces; it leads to the notion of mappings and gives, in particular for Banach spaces, a very satisfactory theory (cf. Chap. VIII of [3]).
It is well known that in this classical theory the notions of differentiability and derivative remain unchanged if one replaces the given norms by equivalent ones, i.e. by norms inducing the same topologies. It is natural therefore to look for a theory which does not use the norms, but only the topologies of the considered vector spaces. In fact, throwing out something which is irrelevant usually leads to a clarification and simplification on one side, and allows a more general theory on the other side. In the case of calculus, such a generalization is indeed desirable in view of applications to certain function spaces which have a natural topology, but no natural norm.