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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

and related areas of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 “size� ...

subsets. These spaces were introduced by Alexander Grothendieck.

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 v ...

with continuous dual

X^

, is called a Schwartz space if it satisfies any of the following equivalent conditions: #For every

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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

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 In functional analysis, a topological vector space (TVS) is said to be quasi-complete or boundedly complete if every closed and bounded subset is complete. This concept is of considerable importance for non- metrizable TVSs. Properties * Eve ...

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 space, barrelled topo ...

. The

strong dual space In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...

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 In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spac ...

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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...

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 In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...

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 th ...

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

References

Bibliography