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.
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3 Answers
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There is a book by Sadayuki Yamamuro on "Differential calculus in topological linear spaces." It appeared in the Springer Lecture Notes.
answered Jun 8, 2014 at 5:27
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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
answered Jun 8, 2014 at 7:47
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$\begingroup$ How would the second order Gâteaux derivative be defined? $\endgroup$ Jun 8, 2014 at 15:15
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$\begingroup$ Dear user51761, if that's even your real name ;-), the Wikipedia page has some information about that. For more details, the article by Hamilton and the book by Kriegl & Michor are really very highly recommended. $\endgroup$ Jun 8, 2014 at 16:37
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2$\begingroup$ @user51761 You can find Kriegl & Michor's boook here: Michor $\endgroup$ Jun 8, 2014 at 16:47
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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.
