Grothendieck on Topological Vector Spaces In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he 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 gave the least priority to Topological Tensor Products and Nuclear Spaces.
Now, the questions I have are:


*

*What led Grothendieck to 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 of 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.
 A: 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). 
A: There is another branch of the TVS theory, which is not dead at all. It deals with TVS over non-Archimedean fields. See
C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over Non-Archimedean Valued Fields, Cambridge University Press, 2010. doi:10.1017/CBO9780511729959
A: In the theory of Banach spaces there were at least two major developments partly motivated by the work of Grothendieck himself, in particular by the Grothendieck inequality (see http://en.m.wikipedia.org/wiki/Grothendieck_inequality). 
First, this is the theory of absolutely summing operators which was directly motivated by the work of Grothendieck. Second, this is the theory of type and cotype of Banach spaces founded by Maurey and Pisier; this theory studies the properties of Banach spaces from the probabilistic point of view. 
Both theories were extremely influential on the subject of Banach spaces, in particular on their geometry.
A: 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 Fréchet 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 Fréchet 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 Fréchet 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!
A: An important contribution in this general area which came after Grothendieck's work is the development of probability theory on such spaces like $S'$, $D'$, etc. This started in the fifties with the articles of Itô, Gel'fand and Minlos but only reached a mature stage in the thesis by Fernique [1] during the sixties.
[1] Xavier Fernique, Processus linéaires, processus généralisés, Annales de l'institut Fourier, Volume 17 (1967) no. 1 , p. 1-92, doi:10.5802/aif.249
A: 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. 
A: 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 
