In functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by Alexander Grothendieck.
A Hausdorff locally convex space X with continuous dual , X is called a Schwartz space if it satisfies any of the following equivalent conditions:
Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space.
The strong dual space of a complete Schwartz space is an ultrabornological space.
Every infinite-dimensional normed space is not a Schwartz space.
There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.