# 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.

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At least there are still papers on that subject. But Grothendieck would certainly pay no attention to them and hang on his statement from the 60s ;). – Martin Brandenburg Mar 10 '12 at 19:13
I would prefer it if you replaced "Is TVS really dead?" with a more well-defined, and less subjective, question. – Yemon Choi Mar 11 '12 at 8:06
@Yemon Thank you for sensitizing. I have removed the question. Rephrased the original question. – Uday Mar 11 '12 at 8:35
I upvoted Yemon's comment, but this was a mistake. In my opinion, the theory of TVS is indeed dead, and the most part of the guilt for this lies on Alexander Grothedieck. It must have been evident from the very beginning that there is something wrong in this abundance of topologies on the dual space, duality theories, counter-examples, etc. After its birth the theory immediately turned into a long list of counterexamples. The scientific explanation can't be so intricate, knotty, this is an abuse of professional knowledge. – Sergei Akbarov Sep 28 '14 at 6:51
As an illustration: who knows that every Banach space $X$ becomes relexive, $X^{\star\star}=X$, if we endow its dual space $X^\star$ with the compact-open topology? This was found in 1952 by Marianne Smith. When I am telling this to people they are surprised. Formally this is absurd: the simple explanation is less known than the intricate one. A reference for those who find this unexpected: en.wikipedia.org/wiki/Stereotype_space. – Sergei Akbarov Sep 28 '14 at 7:34

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.

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I vaguely recall Rota reporting similar comments, somewhere in the Indiscrete Thoughts book – Yemon Choi Mar 18 '12 at 20:27

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!

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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

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Thanks for the response. It is well known that Weil never liked Grothendieck's approach. Your answer only gives a reason why we should take Grothndieck's words with a grain of salt. It does not show that TVS is not actually dead. – Uday Mar 11 '12 at 5:30
The framing of the issue matters. If TVS is taken to be a theory supporting rigorous quantum field theory, then pronouncing it dead is obviously premature. I was implicitly suggesting that the axiomatic approach, and consideration of the circle of ideas introduced by Laurent Schwartz, might reasonably explain the content of the statement. It doesn't mean, for example, that Fréchet algebras will never have a good theory. – Charles Matthews Mar 11 '12 at 8:21
I originally left this commen on the question but in seems better here. The mentioned article also contains: "For the whole year [1954] he [Grothendieck] tried without success to make headway on the problem of approximation in topological vector spaces, a problem that was resolved only some twenty years later[...]" I am not precisely sure about the timeline, but the statement to Malgrange might be after this failure. Or, he redecided it was not quite so dead in a short time. In any case, in view of this I would even give less weight to this statement. – user9072 Dec 19 '12 at 13:51

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).

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History will never die, before the human race dies! – kjetil b halvorsen Dec 19 '12 at 12:07
For the record, Fuuyama never said that "history is dead". The title of his book, "the end of history and the last man" is a reference to two well-known phrases of Marx and Nietzche respectively. The thesis of that book is not that "history is dead", but something very simple, namely that liberal democracies are much more robust that anyone expected say 50 or 100 years ago, including marxists, fascists, and liberal democrats alike. Indeed, it is not easy to find a country that was a liberal democracy at any point of the last century and is not one now; OTOH, many countries have become so. – Joël Dec 19 '12 at 14:56
However I agree with your larger point, that there are many statements of the form "X is dead", and that very often they don't mean much more that "I, personally, am done with X". One exception: Nietzche's "God is dead", which means something very precise and deep (and I believe, true) about the evolution of the European culture. – Joël Dec 19 '12 at 15:01
@Joel:Thanks for clarifying Fukuyamas title. I thought Fukuyama was also stating if not explicitly, then implicitly that liberal democracies were the endpoint of the evolution of political forms of a state? Quite, except when the "I" is an influential person in the field, remarks such as these are more influential and drive people away. – Mozibur Ullah Dec 21 '12 at 2:11

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.

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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.

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