Infunctional analysis and related areas ofmathematics,Schwartz spaces aretopological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition oftotally bounded subsets. These spaces were introduced byAlexander Grothendieck.

AHausdorfflocally convex spaceX with continuous dual${\displaystyle X^{\mathrm{\prime }}}$,X is called aSchwartz space if it satisfies any of the following equivalent conditions:^[1]

1. For everyclosedconvex balanced neighborhoodU of the origin inX, there exists a neighborhoodV of0 inX such that for all realr > 0,V can be covered by finitely many translates ofrU.
2. Every bounded subset ofX istotally bounded and for everyclosedconvexbalanced neighborhoodU of the origin inX, there exists a neighborhoodV of0 inX such that for all realr > 0, there exists a bounded subsetB ofX such thatV ⊆B +rU.

Everyquasi-complete Schwartz space is asemi-Montel space. EveryFréchet Schwartz space is aMontel space.^[2]

Thestrong dual space of acomplete Schwartz space is anultrabornological space.

• Vector subspace of Schwartz spaces are Schwartz spaces.
• The quotient of a Schwartz space by a closed vector subspace is again a Schwartz space.
• TheCartesian 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 spaceif 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.

Every infinite-dimensionalnormed space isnot a Schwartz space.^[2]

There existFréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are notMontel spaces.^[2]

• Auxiliary normed space
• Montel space – Barrelled space where closed and bounded subsets are compact

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• Functional analysis
• Topological vector spaces

• CS1 French-language sources (fr)