Schwartz space on Euclidean space - Imported Knowlpedia Content

Definition

The Schwartz space $\mathcal S(\mathbb R^n)$ consists of all smooth functions $f:\mathbb R^n\to\mathbb C$ such that, for every pair of multi-indices $\alpha,\beta$ ,

p_{\alpha,\beta}(f) =\sup_{x\in\mathbb R^n}|x^\alpha\partial^\beta f(x)|<\infty.

Here $\partial^\beta$ denotes an iterated partial derivative. The functions $p_{\alpha,\beta}$ are seminorms, and the topology they generate is the Schwartz topology. With this topology, $\mathcal S(\mathbb R^n)$ is a Fréchet space: complete, metrizable, and locally convex.

Convergence and rapid decay

A sequence $f_j$ converges to $f$ in the Schwartz topology exactly when

p_{\alpha,\beta}(f_j-f)\longrightarrow0

for every $\alpha,\beta$ . Thus convergence controls every derivative, uniformly after multiplication by every monomial. This is stronger than uniform convergence of all derivatives and far stronger than pointwise or $L^p$ convergence. The seminorm formulation records both smoothness and rapid decay without choosing a single rate.

Stability properties

Differentiation, multiplication by a polynomial, translation, and modulation act continuously on $\mathcal S(\mathbb R^n)$ . The Fourier transform is a continuous linear bijection of the Schwartz space onto itself with continuous inverse. These properties make it a natural common domain for Euclidean Fourier analysis and differential operators Hörmander, Chapters 1 and 7.

Relation to distributions

A tempered distribution is a continuous linear functional on the topological vector space $\mathcal S(\mathbb R^n)$ . It is not a Schwartz function, and not every algebraic linear functional on the underlying vector space is tempered. Ordinary distributions are instead continuous on compactly supported test functions; tempered distributions form the subclass that also acts continuously under the polynomial growth-and-decay control encoded by the Schwartz topology.

Fréchet, nuclear, and non-Banach features

The Schwartz space is a nuclear Fréchet space. For $n\geq1$ , its standard topology cannot be defined by a single norm, so it is not a Banach space with that topology. Individual weighted derivative bounds look norm-like, but the full topology requires the countable family of all $p_{\alpha,\beta}$ . This distinction is essential when defining continuity of tempered distributions Trèves, Chapters 10 and 51.

Examples

Every smooth compactly supported function is a Schwartz function. The Gaussian $e^{-|x|^2}$ and every polynomial times a Gaussian also belong to $\mathcal S(\mathbb R^n)$ . The function $(1+|x|^2)^{-1}$ is smooth and decays at infinity, but it is not Schwartz: multiplication by a sufficiently high power of a coordinate makes it unbounded.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. Publisher record. Relevant: Chapters 1 and 7.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Publisher record. Relevant: the examples of Fréchet and nuclear function spaces.
Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999. Publisher record. Relevant: Chapter 8 on distributions and Fourier analysis.