Nuclear space

In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck. The topology on nuclear spaces can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold. All finite-dimensional vector spaces are nuclear. There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a "naturally occurring" topological vector space is a Banach space, then there is a good chance that it is nuclear. Distribution (mathematics)#Topology on the space of distributions and Schwartz kernel theorem Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in . We now describe this motivation. For any open subsets and the canonical map is an isomorphism of TVSs (where has the topology of uniform convergence on bounded subsets) and furthermore, both of these spaces are canonically TVS-isomorphic to (where since is nuclear, this tensor product is simultaneously the injective tensor product and projective tensor product). In short, the Schwartz kernel theorem states that: where all of these TVS-isomorphisms are canonical. This result is false if one replaces the space with (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces with the dual of this space. Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space (which is generally considered one of the "nicest" TVSs)? This question led Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product.