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!

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!