functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

and related areas of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Schwartz spaces are

topological vector spaces In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) whose neighborhoods of the origin have a property similar to the definition of

totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...

subsets. These spaces were introduced by

Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...

Definition

A Hausdorff

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

with continuous dual

X^

, is called a Schwartz space if it satisfies any of the following equivalent conditions: #For every closed convex balanced neighborhood of the origin in , there exists a neighborhood of in such that for all real , can be covered by finitely many translates of . #Every bounded subset of is

and for every closed

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

balanced neighborhood of the origin in , there exists a neighborhood of in such that for all real , there exists a bounded subset of such that .

Properties

Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a

Montel space In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector ...

. The strong dual space of a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

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 The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...

is ''not'' a Schwartz space. There exist

Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to ...

s that are not Schwartz spaces and there exist Schwartz spaces that are not

References

Bibliography