Frechet Derivative in General Topological Vector Space If I have a two Hausdorff topological vector spaces, $E$ and $F$ and a mapping $f:E\to F$, is it possible to have a meaningful notion of the derivative of $f$ if the space cannot be endowed with a metric or norm? I have looked at Frechet derivatives, but it seems that they need some notion of a norm, except at $0$, where the notion of the function being tangent at $0$ is based solely on the notion of topological neighborhoods. 
 A: There is a book by Sadayuki Yamamuro on "Differential calculus in topological linear spaces." It appeared in the Springer Lecture Notes.
A: Here's something that's a bit concrete. The Gâteaux derivative is defined as follows:
$$ df(x,h) = \lim_{t\to 0} \frac{f(x+th)-f(x)}{t} , $$
which requires only a topology on the codomain $F$. The trouble is of course that $df(x,-)$ need not be a linear or continuous map. If you require further conditions, such as the joint continuity of $df(-,-)$ as a function of its arguments, then more of these desirable features follow. This is discussed, for instance, in
Hamilton, R. S. (1982), The inverse function theorem of Nash and Moser, Bull. AMS. 7 (1): 65–222, MR 656198
A: The theory of differentiability on topological vectors spaces has a long traditon (the first work that I could quote offhand is by J. Sebastião e Silva (1956) but there are almost certainly earlier examples) and whole libraries have been written about it.  A fairly recent overview can be found in " A convenient setting for global analysis" by Kriegl and Michor.
