Schwartz room (general)

In mathematics, a Schwartz space is a special class of locally convex vector spaces . Many rooms that are important in the applications, e.g. B. Spaces with differentiable functions are Schwartz spaces. The space of functions quickly falling (s. U.) Is in the distribution theory sometimes referred to as the room Schwartz called, although the representative only representative here to be discussed room class. The name Schwartz room (after Laurent Schwartz ) goes back to Alexander Grothendieck . In the literature, the term -space is also common; a complete Schwartz space is then also called a space. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle S}$ ${\displaystyle {\overline {S}}}$

definition

A locally convex space is called a Schwartz space if there is a null neighborhood for every normalized space and every continuous linear operator , so that the image is precompact . ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle A\colon E\rightarrow F}$ ${\displaystyle V\subset E}$ ${\displaystyle A(V)}$

This is the case if and only if there is a null neighborhood for every Banach space and every continuous linear operator such that is compact . ${\displaystyle F}$ ${\displaystyle A:E\rightarrow F}$ ${\displaystyle V\subset E}$ ${\displaystyle {\overline {A(V)}}}$

An internal characterization reads:

A locally convex space is exactly then if for every zero neighborhood Schwartz space, a neighborhood is, one that at any finite number of points to be found. ${\displaystyle E}$ ${\displaystyle U\subset E}$ ${\displaystyle V\subset E}$ ${\displaystyle \epsilon >0}$ ${\displaystyle x_{1},\ldots ,x_{n}\in E}$ ${\displaystyle \textstyle V\subset \bigcup _{j=1}^{n}(x_{j}+\epsilon U)}$

Precompact semi-norms

Schwartz rooms can also be characterized using the constant semi-norms . A semi-norm on a locally convex space is called precompact if there is a null sequence in and an equidistant sequence in the strong dual space , so that the inequality holds for all . (The sequence is called equidistant if there is a continuous semi-norm on with for all and .) ${\displaystyle p}$ ${\displaystyle E}$ ${\displaystyle (\zeta _{n})_{n}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle E\,'}$ ${\displaystyle x\in E}$ ${\displaystyle \textstyle p(x)\leq \sup _{n\in \mathbb {N} }|\zeta _{n}f_{n}(x)|}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle q}$ ${\displaystyle E}$ ${\displaystyle |f_{n}(x)|\leq q(x)}$ ${\displaystyle x\in E}$ ${\displaystyle n\in {\mathbb {N} }}$

Precompact semi-norms are continuous, because the above designations give the estimate . The converse is generally not correct, it rather represents a characterization of the Schwartz spaces, because the following applies: ${\displaystyle \textstyle p(x)\leq \sup _{n\in \mathbb {N} }|\zeta _{n}f_{n}(x)|\leq \sup _{n\in \mathbb {N} }|\zeta _{n}|\cdot q(x)}$

A locally convex space is a Schwartz space if and only if every continuous semi-norm is precompact. ${\displaystyle E}$

Examples

• Among the normalized spaces are precisely the finite-dimensional spaces Schwartz spaces.
• Every complete nuclear space is a Schwartz space.
• Be the space of all functions for which all suprema are finite. The multi-index notation was used. The space with the semi-norms is called the space of rapidly falling functions. It is a Schwartz room and is sometimes referred to as the Schwartz room . ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle f:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }}$ ${\displaystyle \textstyle p_{k,m}(f):=\sup _{|\alpha |\leq k}\sup _{x\in {\mathbb {R} }^{n}}|(1+|x|^{2})^{m}D^{\alpha }f(x)|}$ ${\displaystyle {\mathcal {S}}({\mathbb {R} }^{n})}$ ${\displaystyle \{p_{k,m};\,k,m\in {\mathbb {N} }_{0}\}}$
• Each sequence defines a linear functional on the sequence space of the bounded sequences by fixing . This space is provided with the finest locally convex topology, so that the dual space with regard to this identification coincides with . According to Mackey-Arens' theorem, there is such a topology, the Mackey topology . The locally convex space is a complete Schwartz space that is not nuclear. ${\displaystyle (a_{n})_{n}\in \ell ^{1}}$ ${\displaystyle \textstyle (x_{n})_{n}\mapsto \sum _{n=1}^{\infty }a_{n}x_{n}}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \tau (\ell ^{\infty },\ell ^{1})}$ ${\displaystyle (\ell ^{\infty },\tau (\ell ^{\infty },\ell ^{1}))}$

properties

• Subspaces and quotient spaces after closed subspaces of Schwartz spaces are again Schwartz spaces.
• Any Schwartz room product is again a Schwartz room.
• Complete, quasi-tiled Schwartz rooms are Montel rooms . But there are Fréchet -Montel rooms that are not Schwartz rooms.
• A locally convex space is a Schwartz space if and only if there is a set such that is topologically isomorphic to a subspace of . In this sense, Schwartz is a universal room . ${\displaystyle E}$ ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle (\ell ^{\infty },\tau (\ell ^{\infty },\ell ^{1}))^{I}}$ ${\displaystyle (\ell ^{\infty },\tau (\ell ^{\infty },\ell ^{1}))}$

Complete Schwartz rooms

Complete Schwartz rooms have special properties and allow further characterizations. If a continuous semi-norm is on the locally convex space , then a closed subspace is explained by and by a norm on the factor space . The completion of this standardized space is denoted by. Is a more steady with semi-norm , is defined a steady linear operator , the continuously at a linear operator continue leaves. The names of the local Banach spaces and the operators are called canonical figures of . With these terms, complete Schwartz spaces can be characterized as follows: ${\displaystyle p}$ ${\displaystyle E}$ ${\displaystyle N_{p}:=\{x\in E;p(x)=0\}}$ ${\displaystyle E}$ ${\displaystyle \|x+N_{p}\|_{p}:=p(x)}$ ${\displaystyle E_{p}:=E/N_{p}}$ ${\displaystyle B_{p}}$ ${\displaystyle q}$ ${\displaystyle p\leq q}$ ${\displaystyle x+N_{q}\mapsto x+N_{p}}$ ${\displaystyle E_{q}\rightarrow E_{p}}$ ${\displaystyle \kappa _{qp}:B_{q}\rightarrow B_{p}}$ ${\displaystyle B_{p}}$ ${\displaystyle \kappa _{qp}}$ ${\displaystyle E}$

A locally convex space is a complete Schwartz space if and only if there is another continuous semi-norm for every continuous semi- norm , so that the canonical mapping is a compact operator . ${\displaystyle p}$ ${\displaystyle q\geq p}$ ${\displaystyle \kappa _{qp}:B_{q}\rightarrow B_{p}}$

It is of course sufficient to restrict oneself to a directed system of generating semi-norms.

In complete Schwartz spaces, the Bolzano-Weierstrass theorem applies , that is, a set is compact if and only if it is closed and bounded .

literature

• K. Floret, J. Wloka: Introduction to the theory of locally convex spaces. Lecture Notes in Mathematics 56, 1968.
• HH Schaefer: Topological Vector Spaces. Springer, 1971.
• H. Jarchow: Locally Convex Spaces. Teubner, Stuttgart 1981.
• Yau-Chuen Wong: Introductory Theory of Topological Vector Spaces. Marcel Dekker Ltd., 1992.
• R. Meise, D. Vogt: Introduction to functional analysis. Vieweg, 1992. ISBN 3-528-07262-8