Schwartz topological vector space

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. 
Definition.
A Hausdorff locally convex space with continuous dual formula_1, is called a Schwartz space if it satisfies any of the following equivalent conditions: 
Properties.
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.
Examples and sufficient conditions.
Counter-examples.
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. 