Grothendieck on Topological Vector Spaces - MathOverflow

Grothendieck on Topological Vector Spaces

2012-03-10T18:28:39Z

In the short biography article on the Alexander Grothedieck

http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf ,

it is mentioned that after Grothendieck submitting his first thesis on Topological Vector Spaces(TVS), apparently, told Bernard Malgrange that 'There is nothing more to do, the subject is dead.'

Also, after nearly two decades, while listing 12 topics of his interest Grothendieck gives least priority to Topological Tensor Products and Nuclear Spaces.

Now, the questions I have are:
What made Grothendieck make this pronouncement on TVS? Could somebody indicate some significant problems or contributions in this area after Grothendieck? My interest is not in the applications or the impact on the subject on other areas of mathematics but I am interested in knowing about the growth of TVS theory itself.

Thank you, in advance, for your answer.

Answer by Charles Matthews for Grothendieck on Topological Vector Spaces

2012-03-10T21:52:23Z

It seems clear enough to me that Grothendieck was (perhaps is) sui generis as a mathematician, something that can be said of a few other mathematicians in each of the 19th and 20th centuries (e.g. Ramanujan). There seems to be something in his approach that both leads others to hyperbole about him, and led him to apply hyperbole in his pronouncements on mathematics. Which is not an unmixed blessing: cf. Weil's comments in the preface to Basic Number Theory. This particular pronouncement seems less interesting than others. It is the type of thing that the Bourbaki group often said, and its only justification lies in the need to have some sort of heuristic in choosing a research area. The historical assessment seems to be that distribution theory had raised issues in TVS theory, and Grothendieck dealt with those

Answer by Mozibur Ullah for Grothendieck on Topological Vector Spaces

2012-03-11T02:34:55Z

These kind of statements are made from time, not just within subfields of mathematics, but also within the larger world. From painting is dead (I'm not sure who said this) & history is dead (Fukuyama).

Answer by Ronnie Brown for Grothendieck on Topological Vector Spaces

2012-03-18T14:33:35Z

Grothendieck told me in 1985 (1986?) that he was proud of the fact that his published thesis got a prize as one of the most quoted papers. I just looked it up in MathScNet and it has 335 citations given there. On the other hand he writes that he found in analysis not enough geometry, and relished the wider pastures in algebraic geometry.

Answer by Alexei Pirkovskii for Grothendieck on Topological Vector Spaces

2012-12-19T10:46:35Z

After Grothendieck, a number of significant results in TVS theory was obtained by D.Vogt and his collaborators. I especially like results on "automatic splittng" of exact sequences of Fr\'echet spaces. For example, a theorem by Vogt and Wagner states that a short exact sequence $0\to E\to F\to G\to 0$ of nuclear Fr\'echet spaces splits provided that $E$ has property $(\Omega)$ and $G$ has property $(DN)$ (see, e.g., Meise and Vogt's book "Introduction to Functional Analysis"). How one can apply this result? Suppose, for example, that $V$ is a smooth algebraic subvariety of $\mathbb C^n$, let $\mathcal O(\mathbb C^n)$ denote the algebra of holomorphic functions on $\mathbb C^n$, and let $I\subset\mathcal O(\mathbb C^n)$ be the ideal of functions vanishing on $V$. By Cartan's Theorem B, the sequence $0\to I\to\mathcal O(\mathbb C^n)\to \mathcal O(V)\to 0$ is exact. It is rather easy to show that $I$ has $(\Omega)$, and a deep result of Zaharyuta, Vogt, Aytuna, and Palamodov states that $\mathcal O(V)$ has $(DN)$. Hence the above sequence splits in the category of Fr\'echet spaces.

In fact, there are much more "automatic splitting" results, with numerous applications to Complex Analysis and PDE's. So the subject is still alive!