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:
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What led Grothendieck to make this pronouncement on TVS?
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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.
Best Answer
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!