The Fourier transform of a function in $L^1(\mathbb{R}^n)$ is bounded and continuous, but not necessarily integrable again — as demonstrated by the indicator function example on the [Fourier Transform](/pages/1049) page. The Fourier transform of a function in $L^2(\mathbb{R}^n)$ is defined only via a density argument and has no pointwise formula. Neither class is closed under [differentiation](/page/Derivative) in a way compatible with the Fourier transform: differentiating an $L^1$ function may leave $L^1$, and multiplying by a polynomial (the frequency-side counterpart of differentiation) may leave $L^2$. If we want to differentiate freely, multiply by polynomials freely, and take [Fourier transforms](/page/Fourier%20Transform) freely — all within a single function space — we need something smaller and better behaved than any $L^p$ space, but still large enough to serve as a dense test space for [distributions](/page/Distribution). The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is the canonical answer. It consists of smooth [functions](/page/Function) that, together with all their derivatives, decay faster than any polynomial at infinity. This single condition — rapid decay of all derivatives — is precisely what makes the space closed under both differentiation and Fourier transform, and what makes it dense in every $L^p$ space so that distributions defined on it capture all $L^p$ functions. ## Motivation [motivation] ### Why $L^1$ and $L^2$ Are Not Enough The Fourier transform interacts with two basic calculus operations: differentiation (which on the frequency side becomes multiplication by $\xi$) and multiplication by a polynomial (which on the frequency side becomes differentiation). For the transform to be well-defined as a pointwise integral, the function must be integrable. For the differentiation rule to be applicable, the derivative must also be integrable. For the multiplication-by-$\xi$ rule to hold, the result of multiplying by $\xi$ must remain in a class where the transform is defined. Consider $L^1(\mathbb{R}^n)$. It is closed under neither differentiation nor polynomial multiplication: if $f \in L^1$, the derivative $\partial_{x_j} f$ may not be in $L^1$, and $x_j f$ may not be in $L^1$ either (think of a function with slow polynomial decay). Consider $L^2(\mathbb{R}^n)$. While the [Plancherel Theorem](/theorems/247) extends $\mathcal{F}$ to a unitary map on $L^2$, the extension is abstract — there is no integral formula for $\hat{f}$ when $f \in L^2 \setminus L^1$. Worse, $L^2$ is not closed under polynomial multiplication: if $f \in L^2$, the function $x_j f$ may fail to be square-integrable. What we need is a space of functions that is simultaneously closed under differentiation, closed under polynomial multiplication, and contained in $L^1 \cap L^2$ so that the Fourier transform has a concrete integral formula. The Schwartz space achieves all three. ### The Key Idea: Rapid Decay of All Derivatives The obstruction in $L^1$ and $L^2$ is always the same: functions do not decay fast enough at infinity. The fix is to demand that every derivative of $f$ decays faster than every polynomial. Concretely, we require that for every multi-index $\alpha$ (controlling polynomial growth) and every multi-index $\beta$ (controlling derivatives), the quantity $\sup_{x} |x^\alpha \partial^\beta f(x)|$ is finite. This is the strongest decay condition one can impose while still having a rich supply of functions: every compactly supported smooth function satisfies it (trivially), and so does the Gaussian $e^{-|x|^2}$ (because exponential decay beats any polynomial). The payoff is immediate. If $f$ satisfies the rapid decay condition, then so does $\partial_{x_j} f$ (derivatives of rapidly decaying functions are rapidly decaying), and so does $x_j f$ (multiplying by $x_j$ "uses up" one order of polynomial decay, but there are infinitely many orders to spare). This is the essential algebraic closure that $L^p$ spaces lack. [/motivation] ## The Space and Its Topology Throughout this page, we use standard multi-index notation: for $\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbb{N}_0^n$, we write $x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ for the monomial, $\partial^\alpha = \partial_{x_1}^{\alpha_1} \cdots \partial_{x_n}^{\alpha_n}$ for the partial differential operator, and $|\alpha| = \alpha_1 + \cdots + \alpha_n$ for the order. The Schwartz space is defined by finiteness of a family of semi-norms that simultaneously measure polynomial growth and derivative behaviour. [definition:Schwartz Semi-Norm] For $\alpha, \beta \in \mathbb{N}_0^n$, the $(\alpha, \beta)$-**Schwartz semi-norm** is the map \begin{align*} \|\cdot\|_{\alpha,\beta}: C^\infty(\mathbb{R}^n) &\to [0, \infty] \\ f &\mapsto \sup_{x \in \mathbb{R}^n} |x^\alpha \, \partial^\beta f(x)|. \end{align*} [/definition] The semi-norm $\|f\|_{\alpha,\beta}$ measures how well the derivative $\partial^\beta f$ resists the polynomial weight $x^\alpha$. If $\|f\|_{\alpha,\beta} < \infty$, then $|\partial^\beta f(x)| \leq \|f\|_{\alpha,\beta} / |x^\alpha|$, which forces $\partial^\beta f$ to decay at least as fast as $|x|^{-|\alpha|}$ in the directions where $x^\alpha$ grows. Since we demand this for *every* $\alpha$, the decay must be faster than any polynomial — this is the rapid decay condition. [definition:Schwartz Space] The **Schwartz space** on $\mathbb{R}^n$, denoted $\mathcal{S}(\mathbb{R}^n)$, is the set of all smooth functions for which every Schwartz semi-norm is finite: \begin{align*} \mathcal{S}(\mathbb{R}^n) &:= \left\{ f \in C^\infty(\mathbb{R}^n) \;\middle|\; \|f\|_{\alpha,\beta} < \infty \text{ for every } \alpha, \beta \in \mathbb{N}_0^n \right\}. \end{align*} [/definition] [definition:Schwartz Topology] The **Schwartz topology** on $\mathcal{S}(\mathbb{R}^n)$ is the coarsest [locally convex](/page/Topological%20Vector%20Space) topology making every semi-norm $\|\cdot\|_{\alpha,\beta}$ continuous. [/definition] The [Neighbourhood Base of the Seminorm-Generated Locally Convex Topology](/theorems/664) guarantees that this topology exists, is unique, and has an explicit neighbourhood base: a base of neighbourhoods of any $f \in \mathcal{S}(\mathbb{R}^n)$ consists of all [sets](/page/Set) of the form $\bigcap_{i=1}^m \{g \in \mathcal{S}(\mathbb{R}^n) : \|f - g\|_{\alpha_i, \beta_i} < \varepsilon_i\}$ for finitely many pairs $(\alpha_i, \beta_i) \in \mathbb{N}_0^n \times \mathbb{N}_0^n$ and positive reals $\varepsilon_i > 0$. The indexing set $\mathbb{N}_0^n \times \mathbb{N}_0^n$ is countable, which has an important consequence for metrisability (see below). The following result characterises sequential convergence in the Schwartz topology. It is what one uses in practice when verifying that a sequence converges. [quotetheorem:252] The forward direction uses the neighbourhood base: for fixed $\alpha, \beta$ and $\varepsilon > 0$, the set $\{g : \|f - g\|_{\alpha,\beta} < \varepsilon\}$ is a basic neighbourhood of $f$ (a single seminorm ball), so topological convergence forces $f_k$ into this set eventually. The converse uses the finite-intersection structure: every neighbourhood of $f$ contains a finite intersection of semi-norm balls, and the hypothesis $\|f_k - f\|_{\alpha,\beta} \to 0$ for every $(\alpha, \beta)$ provides eventual containment in each factor, so a single threshold handles the entire intersection. ### The Schwartz Space as a Fréchet Space The family of semi-norms generating the Schwartz topology is countable (indexed by $\mathbb{N}_0^n \times \mathbb{N}_0^n$). A locally convex topology generated by countably many semi-norms is metrisable: a compatible metric is \begin{align*} d(f,g) &= \sum_{\alpha, \beta \in \mathbb{N}_0^n} 2^{-|\alpha|-|\beta|} \frac{\|f - g\|_{\alpha,\beta}}{1 + \|f - g\|_{\alpha,\beta}}, \end{align*} where the sum is taken over any enumeration of the [countable set](/page/Countable%20Set) $\mathbb{N}_0^n \times \mathbb{N}_0^n$. Each summand is bounded by $2^{-|\alpha|-|\beta|}$, so the [series](/page/Series) converges. The metric $d$ induces the same topology as the semi-norm family (this follows from the general theory of locally convex spaces with countably many semi-norms). The resulting [metric space](/page/Metric%20Space) is complete: [quotetheorem:253] The proof extracts a smooth [limit](/page/Limit) from [uniform convergence](/page/Uniform%20Convergence) of all derivatives (which is the content of the Cauchy condition in $\|\cdot\|_{0,\beta}$), then verifies rapid decay by passing to the limit pointwise in the general Cauchy condition. Since the Schwartz topology is metrizable, locally convex, and complete, $\mathcal{S}(\mathbb{R}^n)$ is a [Fréchet space](/page/Fr%C3%A9chet%20Space). Convergence in $\mathcal{S}(\mathbb{R}^n)$ is therefore much stronger than convergence in any single $L^p$ space: it requires *every* derivative of $f_k$ to converge *uniformly against every polynomial weight*. A [sequence](/page/Sequence) can converge to zero in $L^2$ while diverging in $\mathcal{S}$ — for instance, a bump function of fixed shape sliding to infinity loses no $L^2$ mass but its polynomial-weighted semi-norms blow up (see Problem 2). ### Algebraic Closure Properties The Schwartz space is closed under the two operations that $L^p$ spaces fail to be closed under: differentiation and polynomial multiplication. These closures are direct consequences of the semi-norm definition and are the reason $\mathcal{S}(\mathbb{R}^n)$ is invariant under the Fourier transform. [quotetheorem:454] Part (1) is immediate from the semi-norm definition: $\|\partial^\gamma f\|_{\alpha,\beta} = \|f\|_{\alpha, \beta + \gamma}$, so each output semi-norm equals a single input semi-norm. Part (2) requires the Leibniz rule: expanding $\partial^\beta(x^\gamma f)$ via the Leibniz rule produces finitely many terms, each of which is a polynomial in $x$ times a derivative of $f$. The polynomial factors are absorbed by the rapid decay of $f$, and each resulting term is bounded by finitely many Schwartz semi-norms of $f$. These two closure properties are precisely dual under the Fourier transform: differentiation on the spatial side corresponds to polynomial multiplication on the frequency side (the [Fourier Differentiation Rule](/theorems/249)), and polynomial multiplication on the spatial side corresponds to differentiation on the frequency side. Among the function spaces $L^p$, $C^k$, and $C_0$, none is simultaneously closed under both operations; the Schwartz space is the intersection of the kernels of all the obstructions. ## Examples [example:Gaussian In Schwartz Space] The standard Gaussian is the prototypical Schwartz function. Define \begin{align*} g: \mathbb{R}^n &\to \mathbb{R} \\ x &\mapsto e^{-|x|^2}. \end{align*} For any multi-index $\beta$, the partial derivative $\partial^\beta g(x)$ is a polynomial in $x_1, \ldots, x_n$ multiplied by $e^{-|x|^2}$. This follows by induction: $\partial_{x_j} e^{-|x|^2} = -2x_j e^{-|x|^2}$, and each subsequent differentiation produces another polynomial factor times $e^{-|x|^2}$. Therefore, for any $\alpha, \beta \in \mathbb{N}_0^n$, \begin{align*} |x^\alpha \, \partial^\beta g(x)| &\leq |p_{\alpha,\beta}(x)| \, e^{-|x|^2} \end{align*} where $p_{\alpha,\beta}$ is a polynomial whose degree and coefficients depend on $\alpha$ and $\beta$. Since exponential decay dominates any polynomial growth — $\lim_{|x| \to \infty} |p(x)| e^{-|x|^2} = 0$ for every polynomial $p$ — each of these expressions is bounded over $\mathbb{R}^n$, giving $\|g\|_{\alpha,\beta} < \infty$. Hence $g \in \mathcal{S}(\mathbb{R}^n)$. As computed on the [Fourier Transform](/pages/1049) page, the Fourier transform of $e^{-|x|^2/2}$ is $(2\pi)^{n/2} e^{-|\xi|^2/2}$ — again a Gaussian. This confirms that $\mathcal{S}(\mathbb{R}^n)$ is non-trivially closed under $\mathcal{F}$. [/example] [example:Compactly Supported Smooth Functions] The space $\mathcal{D}(\mathbb{R}^n)$ of smooth functions with compact support satisfies \begin{align*} \mathcal{D}(\mathbb{R}^n) &\subseteq \mathcal{S}(\mathbb{R}^n). \end{align*} For $f \in \mathcal{D}(\mathbb{R}^n)$ supported in a ball $B(0, R)$, every derivative $\partial^\beta f$ is also supported in $B(0, R)$, so $x^\alpha \partial^\beta f$ is a continuous function vanishing identically outside $B(0, R)$, hence bounded. This gives $\|f\|_{\alpha,\beta} < \infty$ for all $\alpha, \beta$. The inclusion is strict: the Gaussian $e^{-|x|^2}$ belongs to $\mathcal{S}(\mathbb{R}^n)$ but is everywhere positive, so it cannot have compact support. Thus $\mathcal{D}(\mathbb{R}^n) \subsetneq \mathcal{S}(\mathbb{R}^n)$. Despite being a proper subset, $\mathcal{D}(\mathbb{R}^n)$ is dense in $\mathcal{S}(\mathbb{R}^n)$: [quotetheorem:455] The proof is a truncation argument: multiply $f \in \mathcal{S}$ by smooth cutoffs $\chi_k$ that equal $1$ on $B(0,k)$ and have support in $B(0, 2k)$. The products $\chi_k f$ belong to $\mathcal{D}(\mathbb{R}^n)$, and the Leibniz rule together with the rapid decay of $f$ ensures that $\|\chi_k f - f\|_{\alpha,\beta} \to 0$ for every $\alpha, \beta$. This density is used in the relationship between $\mathcal{D}'(\mathbb{R}^n)$ and the space of [tempered distributions](/page/Tempered%20Distributions) $\mathcal{S}'(\mathbb{R}^n)$: it implies that the restriction map $\mathcal{S}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)$ is injective (a continuous linear functional on $\mathcal{S}$ is determined by its values on the dense subspace $\mathcal{D}$). [/example] ### What Can Go Wrong: Failure of Schwartz Conditions Not every smooth, rapidly decaying function lies in $\mathcal{S}(\mathbb{R}^n)$. The failure can occur in two ways: lack of smoothness, or rapid growth of derivatives despite rapid decay of the function itself. [example:Failure Of Smoothness] In dimension $n = 1$, define \begin{align*} h: \mathbb{R} &\to \mathbb{R} \\ x &\mapsto e^{-|x|}. \end{align*} The function $h$ decays exponentially at infinity, so the zero-derivative semi-norms $\|h\|_{\alpha, 0}$ are all finite. But $h$ has a corner at the origin — $h'(0^+) = -1$ and $h'(0^-) = 1$ — so $h \notin C^1(\mathbb{R})$, let alone $C^\infty(\mathbb{R})$. The smoothness requirement is violated at a single point. [/example] [example:Failure Of Derivative Decay] A subtler failure occurs when the function itself is smooth and rapidly decaying, but its derivatives are not. In dimension $n = 1$, define \begin{align*} \phi: \mathbb{R} &\to \mathbb{R} \\ x &\mapsto e^{-x^2} \sin(e^{x^2}). \end{align*} The function $\phi$ is smooth (it is a composition of smooth functions) and decays rapidly: $|\phi(x)| \leq e^{-x^2} \to 0$ faster than any polynomial. However, differentiating produces \begin{align*} \phi'(x) &= -2x \, e^{-x^2} \sin(e^{x^2}) + 2x \, \cos(e^{x^2}), \end{align*} and the second term $2x \cos(e^{x^2})$ grows linearly in $|x|$ and oscillates without decay. For any $\alpha \geq 2$, the semi-norm $\|\phi\|_{(\alpha),(1)} = \sup_x |x^\alpha \phi'(x)|$ is infinite because the term $|x^{\alpha+1} \cos(e^{x^2})|$ is unbounded. Hence $\phi \notin \mathcal{S}(\mathbb{R})$ despite being smooth and rapidly decaying. [/example] ## Fourier Transform and Density The two most important structural properties of $\mathcal{S}(\mathbb{R}^n)$ for analysis and PDE theory are closure under the Fourier transform and density in the $L^p$ spaces. Together, these make $\mathcal{S}(\mathbb{R}^n)$ the right domain on which to define the Fourier transform before extending to $L^2$ (via the [Plancherel Theorem](/theorems/247)) or to tempered distributions. ### Fourier Invariance [quotetheorem:228] The proof uses the algebraic closure properties established above together with the two exchange identities: $\mathcal{F}(\partial_{x_j} f)(\xi) = i\xi_j \, \mathcal{F}f(\xi)$ converts spatial derivatives into frequency-side polynomial factors, while $\partial_{\xi_j}(\mathcal{F}f)(\xi) = -i \, \mathcal{F}(x_j f)(\xi)$ converts spatial polynomial weights into frequency-side derivatives. These identities are iterated to express every Schwartz semi-norm of $\mathcal{F}f$ as the $L^\infty$ norm of a single Fourier transform, which is then bounded by an $L^1$ norm, and the $L^1$ norm is in turn bounded by finitely many Schwartz semi-norms of $f$. This gives both the mapping property $\mathcal{F}: \mathcal{S} \to \mathcal{S}$ and [continuity](/page/Continuity). The same argument applies to $\mathcal{F}^{-1}$ (whose exchange identities differ only by signs), establishing that $\mathcal{F}^{-1}$ is also a continuous map $\mathcal{S} \to \mathcal{S}$. Bijectivity follows from the [Fourier Inversion Theorem](/theorems/246): $\mathcal{F}^{-1} \circ \mathcal{F} = \mathrm{id}$ on $\mathcal{S}$. ### Continuous Embedding and Density in $L^p$ The inclusion $\mathcal{S}(\mathbb{R}^n) \subseteq L^p(\mathbb{R}^n)$ is not merely set-theoretic — it is continuous, meaning that $L^p$ convergence is controlled by finitely many Schwartz semi-norms: [quotetheorem:254] The key estimate is that every Schwartz function satisfies the pointwise bound $|f(x)| \leq C(1 + |x|)^{-N} \max_{|\alpha| \leq N} \|f\|_{\alpha, 0}$ for any chosen $N$: the rapid decay built into the semi-norm conditions directly controls the $L^p$ norm via the [integrability](/page/Integral) of $(1+|x|)^{-Np}$ when $Np > n$. [quotetheorem:229] The density result is what allows arguments developed on $\mathcal{S}(\mathbb{R}^n)$ to be extended to all of $L^p$. For instance, the Plancherel isometry is first established on $\mathcal{S}(\mathbb{R}^n)$ (where both sides of $\|\hat{f}\|_{L^2}^2 = (2\pi)^n \|f\|_{L^2}^2$ are well-defined integrals) and then extended to $L^2$ by density and continuity of the embedding. The proof proceeds by showing that $\mathcal{D}(\mathbb{R}^n) \subseteq \mathcal{S}(\mathbb{R}^n)$ and that $\mathcal{D}(\mathbb{R}^n)$ is dense in $L^p$ via [mollification](/page/Standard%20Mollifier). Since $\mathcal{D} \subseteq \mathcal{S} \subseteq L^p$ and $\mathcal{D}$ is dense in $L^p$, the intermediate space $\mathcal{S}$ is dense as well. ### The Schwartz Space Is Not Dense in $L^\infty$ The density result is restricted to $p < \infty$. The space $\mathcal{S}(\mathbb{R}^n)$ is *not* dense in $L^\infty(\mathbb{R}^n)$. The obstruction is that every Schwartz function vanishes at infinity: if $f \in \mathcal{S}(\mathbb{R}^n)$, then for every $N \geq 1$ there exists $C_N > 0$ such that $|f(x)| \leq C_N (1 + |x|)^{-N}$ for all $x$ (this is the pointwise decay bound from the [Continuous Embedding](/theorems/254) with $\beta = 0$), so $|f(x)| \to 0$ as $|x| \to \infty$. Therefore $\mathcal{S}(\mathbb{R}^n) \subseteq C_0(\mathbb{R}^n)$, where $C_0(\mathbb{R}^n)$ denotes the space of continuous functions vanishing at infinity. The constant function $\mathbb{1}$ belongs to $L^\infty(\mathbb{R}^n)$ but satisfies $\|\mathbb{1} - \phi\|_{L^\infty} \geq 1/2$ for every $\phi \in \mathcal{S}(\mathbb{R}^n)$: for any such $\phi$, there exists $R > 0$ with $|\phi(x)| < 1/2$ for all $|x| > R$ (since $\phi$ vanishes at infinity), so $|\mathbb{1}(x) - \phi(x)| > 1/2$ for all $|x| > R$. The closure of $\mathcal{S}(\mathbb{R}^n)$ in $L^\infty$ is contained in $C_0(\mathbb{R}^n)$, which is a proper closed subspace of $L^\infty$. ## The Role of the Schwartz Space in the Broader Theory The Schwartz space sits at a critical juncture in the hierarchy of function and distribution spaces on $\mathbb{R}^n$. Its position is described by the chain of continuous [linear maps](/page/Linear%20Map): \begin{align*} \mathcal{D}(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n) \xrightarrow{\; f \,\mapsto\, T_f \;} \mathcal{S}'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n), \end{align*} where $1 \leq p \leq \infty$. As with the analogous chain on the [Distribution](/pages/1087) page, the nature of each arrow must be specified precisely. The first two arrows are genuine set-theoretic inclusions with continuous injection: every [test function](/page/Test%20Function) is a Schwartz function (compactly supported smooth functions satisfy every rapid-decay condition trivially), and every Schwartz function belongs to $L^p$ (the [Continuous Embedding](/theorems/254) provides the explicit semi-norm bound). Both $\mathcal{D}$ and $\mathcal{S}$ are dense in $L^p$ for $p < \infty$ ([quotetheorem:229]). The third arrow is the canonical embedding $f \mapsto T_f$, which sends an equivalence class $f \in L^p(\mathbb{R}^n)$ to the continuous linear functional $T_f(\varphi) := \int f\varphi \, d\mathcal{L}^n$ on $\mathcal{S}(\mathbb{R}^n)$. This map is linear, injective (by the [Injectivity of the Canonical Embedding](/theorems/450), proved for $L^1_\mathrm{loc}$ and hence for $L^p \subseteq L^1_\mathrm{loc}$), and continuous. It is *not* a set-theoretic inclusion: $f$ is an equivalence class of measurable functions, while $T_f$ is a continuous linear functional. The fourth arrow is restriction of domain: every $T \in \mathcal{S}'(\mathbb{R}^n)$ (a continuous linear functional on $\mathcal{S}(\mathbb{R}^n)$) restricts to a continuous linear functional on $\mathcal{D}(\mathbb{R}^n)$, since $\mathcal{D}(\mathbb{R}^n) \subseteq \mathcal{S}(\mathbb{R}^n)$ and the Schwartz topology (restricted to $\mathcal{D}(\mathbb{R}^n)$) is coarser than the inductive limit topology. This restriction map is injective: if $T|_\mathcal{D} = 0$, then $T$ vanishes on a dense subspace of $\mathcal{S}$ (by the [Density of Test Functions](/theorems/455)), so $T = 0$ by continuity. The dual space $\mathcal{S}'(\mathbb{R}^n)$ — the space of [tempered distributions](/pages/1053) — is the largest space on which the Fourier transform can be defined by duality, via the [Fourier Transform as Automorphism of Tempered Distributions](/theorems/230). The "temperedness" condition (continuity with respect to the Schwartz seminorms) excludes distributions of super-polynomial growth. For instance, the [regular distribution](/page/Regular%20Distribution) $T_g$ with $g(x) = e^{e^x}$ defines a continuous linear functional on $\mathcal{D}(\mathbb{R})$ but not on $\mathcal{S}(\mathbb{R})$: the pairing $\int e^{e^x} \varphi(x) \, d\mathcal{L}^1(x)$ is finite for $\varphi \in \mathcal{D}(\mathbb{R})$ (compact support kills the growth) but may diverge for $\varphi \in \mathcal{S}(\mathbb{R})$ (polynomial decay cannot compensate doubly-exponential growth). Thus $T_g \in \mathcal{D}'(\mathbb{R}) \setminus \mathcal{S}'(\mathbb{R})$. The dual pairing reverses the inclusion order: as the test function spaces shrink ($\mathcal{D} \hookrightarrow \mathcal{S}$), the distribution spaces grow ($\mathcal{S}' \hookrightarrow \mathcal{D}'$). A functional on a smaller space of test functions needs to satisfy fewer continuity conditions, so the corresponding dual space is larger and accommodates wilder objects. For [Sobolev space](/pages/1018) theory, the Schwartz space provides the natural domain for the Fourier-analytic definition of Sobolev norms. The [Sobolev space](/page/Sobolev%20Space) $H^s(\mathbb{R}^n)$ for $s \in \mathbb{R}$ can be defined as the completion of $\mathcal{S}(\mathbb{R}^n)$ under the norm $\|f\|_{H^s} = \|(1 + |\xi|^2)^{s/2} \hat{f}\|_{L^2}$. This definition requires both the Fourier transform (defined on $\mathcal{S}$ as a [topological](/page/Topology) automorphism by [quotetheorem:228]) and the density of $\mathcal{S}$ in $L^2$ (to ensure the completion captures the correct space when $s = 0$). The Schwartz space also appears in the characterisation of Sobolev regularity via the [Helmholtz Operator](/theorems/227): the isomorphism $(1 - \Delta): H^{s+2} \to H^s$ is defined distributionally via the Fourier multiplier $(1 + |\xi|^2)$, and the well-definedness of this multiplier on $\hat{f}$ for $f \in \mathcal{S}$ is what bootstraps the definition to general Sobolev functions. ## References 1. L. Grafakos, *Classical Fourier Analysis*, 3rd ed. (2014). 2. L. Hörmander, *The Analysis of Linear Partial Differential Operators I* (1983). 3. E. M. Stein and R. Shakarchi, *Fourier Analysis: An Introduction* (2003). 4. M. Reed and B. Simon, *Methods of Modern Mathematical Physics I: Functional Analysis* (1980).