Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map $$T\colon C^{-\infty}(X\times Y)\to Bil(C^\infty(X),C^\infty(Y)),$$ where the target is the space of continuous bilinear functionals $C^\infty(X)\times C^\infty(Y)\to \mathbb{C}$. By definition $(T\Phi)(f,g)=\Phi(f\otimes g)$.

It is well known that $T$ is isomorphism of vector spaces. For which standard topologies on the source and the target $T$ is an isomorphism of topological vector spaces?

For example let us choose on $C^{-\infty}(X\times Y)$ the strong topology, and on the target the topology given by seminorms $$||B||_{K,L}:=\sup_{k\in K,l\in L}|B(k,l))|,$$ where $K\subset C^\infty(X), L\subset C^\infty(Y)$ are arbitrary bounded subset. Then $T$ will be a continuous map for these topologies. But is $T$ a topological isomoprhism?

  • $\begingroup$ This true and one of the manifold formulations of the celecbrated kernel $\endgroup$ – priel Jul 21 '15 at 8:03
  • $\begingroup$ celebrated kernel theorem of L. Schwartz. Primary references: Grothendieck's thesis and Schwartz' articles on vector-vaued distributions and smooth functions which are easy to trace. Secondary: the books of F. Treves. $\endgroup$ – priel Jul 21 '15 at 8:10

$T$ is indeed a topological isomorphism. Indepently of the concrete situation, this can be shown by a suitable version of the closed graph theorem, e.g. the one of de Wilde: the domain has to be ultrabornological an the range should have a web (see, e.g., the book Introduction to Functional Analysis of Meise and Vogt, chapter 24). Here $T$ acts from the dual of the nuclear Frechet space $C^\infty(X\times Y)$ to $Bil(C^\infty(X),C^\infty(Y))\cong L(C^\infty(X),C^{-\infty}(Y))$. It follows e.g. from Grothendieck's work that both spaces are ultrabornological, and both spaces also have web. Therefore, one can apply the closed graph theorem fot $T$ and its inverse. (One can even show that the domain and range space are both isomorphic to $s'$, the space of slowly increasing sequences.)

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