title: Schwartz topological vector space
text: # 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 X with continuous dual $X^{\prime }$, X is called a Schwartz space if it satisfies any of the following equivalent conditions:
  
  * For every closed convex balanced neighborhood U of the origin in X, there exists a neighborhood V of 0 in X such that for all real r > 0, V can be covered by finitely many translates of rU.
  * Every bounded subset of X is totally bounded and for every closed convex balanced neighborhood U of the origin in X, there exists a neighborhood V of 0 in X such that for all real r > 0, there exists a bounded subset B of X such that V ⊆ B + rU.
  
  ## 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
  
  * Vector subspace of Schwartz spaces are Schwartz spaces.
  * The quotient of a Schwartz space by a closed vector subspace is again a Schwartz space.
  * The Cartesian product of any family of Schwartz spaces is again a Schwartz space.
  * The weak topology induced on a vector space by a family of linear maps valued in Schwartz spaces is a Schwartz space if the weak topology is Hausdorff.
  * The locally convex strict inductive limit of any countable sequence of Schwartz spaces (with each space TVS-embedded in the next space) is again a Schwartz space.
  
  ### 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.
  
  