Nuclear space

Summary

A nuclear space is a locally convex space whose seminorms can be compared through nuclear operators. These are maps given by absolutely summable series of rank-one maps. Nuclear spaces were introduced by Alexander Grothendieck in his work on topological tensor products. They behave in many ways like finite-dimensional spaces: tensor products are well behaved, and the Schwartz kernel theorem holds for them. The Schwartz space and spaces of smooth functions on open sets of $\R^n$ are nuclear, while no infinite-dimensional normed space is.