The Hilbert tensor product and inductive limits

Abstract

In the present paper we prove that given two inductive limits of Hilbert spaces \(E= \mathrm{ind}_{n\in \mathbb {N}}E_n\) and \(F=\mathrm{ind}_{n\in \mathbb {N}} F_n\) the complete Hilbert tensor product \(E \tilde{\otimes }_{\sigma } F\) of E and F is topologically isomorphic to the inductive limit of the inductive spectrum \(\left( E_n\tilde{\otimes }_{\sigma }F_n\right) _{n\in \mathbb {N}}\). To this end we consider the Hilbert tensor product for the tensor product of spaces equipped with Hilbertian semi norms, spaces, that we call semi-unitary. We conclude with two consequences, first the positive solution of Grothendieck’s probleme des topologies for Frechet–Hilbert spaces and the complete Hilbert tensor product and second the computation of tensor products where at least one space is not Schwartz, e.g. the tensor product of the space of Schwartz distributions \(\fancyscript{D}'\left( \varOmega \right) \) with the space \(\fancyscript{D}_{\mathrm{L}_2}\) of all smooth functions all the derivatives of which are square integrable or its strong dual, which has its application in parameter dependence problems.