Test-Function Spaces, Distributions, Support, and Convergence

A distribution is not defined by a value at each point. It is defined by the number it assigns to every smooth, compactly supported test function. The choice of test-function space supplies the topology that makes “continuous linear functional” meaningful; it also determines what convergence, support, and later operations mean. This replaces informal expressions such as “zero except at one point and infinite there” by exact pairings.

This page works on an open set $\Omega\subseteq\mathbb R^d$ with complex-valued test functions. The pairing is linear in the test function; no complex conjugation is implicit. Tempered distributions, weak derivatives, pullbacks, products, and distributions on manifolds are developed on their own pages.

The test-function space

For a function $\varphi:\Omega\to\mathbb C$, its support is

$$\operatorname{supp}\varphi = \overline{\{x\in\Omega:\varphi(x)\neq0\}},$$

where the closure is taken in $\Omega$. The standard test-function space is

$$\mathcal D(\Omega) = C_c^\infty(\Omega).$$

Thus $\varphi\in\mathcal D(\Omega)$ is smooth and has support contained in some compact set $K\Subset\Omega$. Compact support removes boundary terms and behavior at infinity from local integrations by parts. Infinite differentiability allows derivatives of any finite order to be transferred onto the test function.

The algebraic vector space is not enough; its convergence rule matters. A sequence $\varphi_n$ converges to $\varphi$ in $\mathcal D(\Omega)$ precisely when:

1. there is one compact set $K\Subset\Omega$ containing $\operatorname{supp}\varphi$ and every $\operatorname{supp}\varphi_n$; and

2. for every multi-index $\alpha$,

   $$\sup_{x\in K} \left| \partial^\alpha(\varphi_n-\varphi)(x) \right| \longrightarrow0.$$

The common compact set is not optional. A bump translated farther and farther away converges pointwise to zero, as do all its derivatives at each fixed point, but it does not converge in $\mathcal D(\mathbb R^d)$ because its supports do not remain in one compact set. Conversely, convergence in $\mathcal D$ controls every derivative, not only the function itself. Dyatlov 2022, Chapter 2, §§ 2.1–2.2, PDF states this sequential convergence rule and the equivalent continuity bounds used below.

A concrete nonzero test function is the standard bump

$$b(x) = \begin{cases} \exp\!\left(-\dfrac{1}{1-|x|^2}\right), & |x|<1,\\[6pt] 0, & |x|\geq1. \end{cases}$$

Every derivative approaches zero at $|x|=1$, so the two formulas join smoothly. After normalization,

$$\rho(x) = \frac{b(x)} {\displaystyle\int_{\mathbb R^d}b(y)\,\mathrm d^d y},$$

$\rho$ is nonnegative, supported in the unit ball, and has integral $1$. Rescaled copies of this bump will give the smallest example of distributional convergence.

Distributions are continuous linear functionals

A distribution on $\Omega$ is a continuous linear map

$$u:\mathcal D(\Omega)\longrightarrow\mathbb C, \qquad \varphi\longmapsto \langle u,\varphi\rangle.$$

The space of distributions is denoted $\mathcal D'(\Omega)$. Continuity can be written without invoking abstract topological-vector-space language: for every compact $K\Subset\Omega$, there are constants $C_K>0$ and a nonnegative integer $N_K$ such that

$$|\langle u,\varphi\rangle| \leq C_K \sum_{|\alpha|\leq N_K} \sup_{x\in K}|\partial^\alpha\varphi(x)|$$

whenever $\varphi\in\mathcal D(\Omega)$ and $\operatorname{supp}\varphi\subseteq K$. The number of derivatives needed may depend on $K$. Equivalently, if $\varphi_n\to0$ in $\mathcal D(\Omega)$, then $\langle u,\varphi_n\rangle\to0$.

This continuity condition distinguishes the continuous dual $\mathcal D'(\Omega)$ from the much larger algebraic dual. It is what makes limits and local estimates stable. The definition and local finite-order bound follow Dyatlov 2022, § 2.1, PDF and Duistermaat and Kolk 2010, Chapters 2–3.

Every locally integrable function $f\in L^1_{\mathrm{loc}}(\Omega)$ defines a regular distribution

$$\langle u_f,\varphi\rangle = \int_\Omega f(x)\varphi(x)\,\mathrm d^d x.$$

Indeed, for $\operatorname{supp}\varphi\subseteq K$,

$$|\langle u_f,\varphi\rangle| \leq \left(\int_K|f(x)|\,\mathrm d^d x\right) \sup_{x\in K}|\varphi(x)|.$$

If two locally integrable functions give the same pairing against every test function, they are equal almost everywhere. It is therefore customary to write $f$ for both the function class and its associated distribution.

Not every distribution comes from a locally integrable function. For $a\in\Omega$, the Dirac distribution is

$$\langle\delta_a,\varphi\rangle=\varphi(a).$$

Evaluation is linear and satisfies the continuity bound with $N_K=0$ when $a\in K$. No locally integrable function supported only at $\{a\}$ can have this action: changing a function on a measure-zero set does not change its integral. The symbol $\delta_a(x)$ is therefore kernel notation for the functional above, not a function with a pointwise infinite value. Its derivatives, Jacobian rules, and constraint-surface versions are developed in Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards.

Restriction, vanishing, and support

If $V\subseteq\Omega$ is open, a test function in $\mathcal D(V)$ extends by zero to an element of $\mathcal D(\Omega)$ because its support stays a positive distance from $\partial V$. The restriction of $u\in\mathcal D'(\Omega)$ is consequently defined by

$$\langle u|_V,\varphi\rangle = \langle u,\widetilde\varphi\rangle, \qquad \varphi\in\mathcal D(V),$$

where $\widetilde\varphi$ denotes the zero extension.

This open-set restriction is always defined. Restriction to a lower-dimensional submanifold is instead a pullback and may fail when the distribution is singular in a normal direction; its general criterion is deferred to Singular Support and Wavefront Sets.

The distribution $u$ vanishes on $V$ when $u|_V=0$, or equivalently when it pairs to zero with every test function supported in $V$. Its support is the complement of the largest open set on which it vanishes:

$$\operatorname{supp}u = \Omega\setminus \bigcup_{\substack{V\subseteq\Omega\ \mathrm{open}\\u|_V=0}}V.$$

This definition is local and never asks for $u(x)$. It gives a relatively closed subset of $\Omega$. If the support of $\varphi$ is disjoint from the support of $u$, then

$$\langle u,\varphi\rangle=0.$$

For a continuous regular distribution, this agrees with the usual closure of the nonzero set; for a general $L^1_{\mathrm{loc}}$ function it gives the essential support. In particular,

$$\operatorname{supp}\delta_a=\{a\}.$$

Multiplication by a smooth function is always defined by

$$\langle gu,\varphi\rangle = \langle u,g\varphi\rangle, \qquad g\in C^\infty(\Omega),$$

and satisfies

$$\operatorname{supp}(gu) \subseteq \operatorname{supp}g\cap\operatorname{supp}u.$$

These localization, multiplication, and support statements are established in Dyatlov 2022, §§ 2.3, 3.2, and 4.1, PDF. They also explain why partitions of unity work for distributions: compatible local restrictions determine one global distribution.

Distributional convergence

A sequence $u_n\in\mathcal D'(\Omega)$ converges to $u\in\mathcal D'(\Omega)$ in the usual weak distributional sense when

$$\langle u_n,\varphi\rangle \longrightarrow \langle u,\varphi\rangle$$

for every fixed $\varphi\in\mathcal D(\Omega)$. This is deliberately much weaker than convergence of test functions. The test function stays fixed while the generalized functions vary.

For the normalized bump above, define

$$\rho_\varepsilon(x) = \varepsilon^{-d}\rho(x/\varepsilon), \qquad \varepsilon>0.$$

When $0\in\Omega$ and $\varepsilon$ is small, each $\rho_\varepsilon$ is a test function and also a regular distribution. For any fixed $\varphi\in\mathcal D(\Omega)$, a change of variables gives

$$\begin{aligned} \langle\rho_\varepsilon,\varphi\rangle &= \int_{\mathbb R^d} \varepsilon^{-d}\rho(x/\varepsilon)\varphi(x)\,\mathrm d^d x\\ &= \int_{\mathbb R^d} \rho(y)\varphi(\varepsilon y)\,\mathrm d^d y\\ &\longrightarrow \varphi(0) = \langle\delta_0,\varphi\rangle. \end{aligned}$$

Hence

$$\rho_\varepsilon\longrightarrow\delta_0 \quad\text{in }\mathcal D'(\Omega).$$

The same family does not converge in $\mathcal D(\Omega)$: its height and derivatives grow as the support shrinks. It also converges pointwise to zero away from the origin, not to a pointwise representative of $\delta_0$. Distributional convergence records the limiting action on probes.

Rapid oscillation supplies a complementary example. On $\mathbb R$,

$$u_n(x)=e^{inx}$$

does not converge pointwise, but integration by parts gives

$$\left| \int_{\mathbb R}e^{inx}\varphi(x)\,\mathrm dx \right| \leq \frac{1}{n} \int_{\mathbb R}|\varphi'(x)|\,\mathrm dx \longrightarrow0.$$

Thus $u_n\to0$ in $\mathcal D'(\mathbb R)$. These examples show why a distributional limit cannot be inferred from pointwise behavior alone.

QFT-facing example: fields and kernels must be smeared

The bounded mathematical statement needed in QFT is that a field is accessed through test functions. Symbolically,

$$\Phi(f) = \int_{\mathbb R^d}\Phi(x)f(x)\,\mathrm d^d x, \qquad f\in\mathcal D(\mathbb R^d).$$

Here $\Phi(x)$ need not be an operator at a point. In an operator-valued-distribution formulation, one specifies a common dense domain $D$ and requires, for every $\Psi_1,\Psi_2\in D$, that

$$f\longmapsto \langle\Psi_1,\Phi(f)\Psi_2\rangle$$

be a scalar distribution. The weak matrix elements are continuous in the test-function topology; this does not assert operator-norm continuity or boundedness of $\Phi(f)$. Invariance of the common domain under the smeared fields is an additional condition and is not established here. A Wightman framework usually replaces $\mathcal D$ by the Schwartz space $\mathcal S$ and adds covariance, spectrum, locality, domain, and vacuum hypotheses. None of those extra axioms follows from the definition on this page.

Similarly, a two-point kernel can be a scalar distribution $G\in\mathcal D'(\mathbb R^d\times\mathbb R^d)$. Its defined object is

$$\langle G,F\rangle, \qquad F\in\mathcal D(\mathbb R^{2d}),$$

not necessarily a number $G(x,y)$ at every pair of points. Factorized probes $F(x,y)=f(x)g(y)$ are useful, but the distribution acts on the full two-variable test space.

Wightman’s historical account, Wightman 1996, “How It Was Learned that Quantized Fields Are Operator-Valued Distributions,” pp. 143–178, explains this QFT transition. The physical meaning of smearing, field domains, and regulated point approximations belongs to Quantum Fields as Operator-Valued Distributions. Tempered growth and Fourier transformation are deferred to Tempered Distributions and Fourier Calculus.

Failure modes

A generalized function is not specified by a graph. Pairings with every test function are the definition. Point notation is shorthand only after its action has been fixed.

Linearity without continuity is insufficient. The algebraic dual contains functionals that do not obey the compact-set derivative bounds and therefore are not distributions.

Pointwise convergence is a different statement. Test-function convergence requires common compact support and convergence of every derivative. Distributional convergence requires convergence of pairings against each fixed test function.

A distribution does not automatically act on every smooth function. The constant function $1$ is not in $\mathcal D(\Omega)$ for a nonempty open $\Omega\subseteq\mathbb R^d$. Extra support or growth hypotheses are needed to enlarge the test space. In particular, a compactly supported distribution does extend canonically to $C^\infty(\Omega)$: multiply the smooth argument by any test-function cutoff equal to $1$ near the distribution’s support.

Support is not singular support. A smooth nonzero function may have support equal to all of $\Omega$ while having no singularities. Directional singular information belongs to the later wavefront-set page.

Two distributions cannot generally be multiplied. Multiplication by a smooth function is defined, but a product such as $\delta_0^2$ or a pointwise field square needs additional criteria or an extension procedure.

Exercises

1. Let $f\in L^1_{\mathrm{loc}}(\Omega)$. Prove directly that $\varphi\mapsto\int f\varphi$ is a distribution.

   Check

   For each $K\Subset\Omega$ and $\operatorname{supp}\varphi\subseteq K$,

   $$\left|\int_\Omega f\varphi\right| \leq \|f\|_{L^1(K)} \sup_K|\varphi|.$$

   This is the required continuity estimate with $N_K=0$. Linearity is immediate.

2. Explain why the normalized functions $\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)$ converge to $\delta_0$ in $\mathcal D'$ but not to zero in $\mathcal D$.

   Check

   Against fixed $\varphi$, substitution $x=\varepsilon y$ gives $\int\rho(y)\varphi(\varepsilon y)\,\mathrm d^d y\to\varphi(0)$. As test functions, however, $\sup|\rho_\varepsilon|=\varepsilon^{-d}\sup|\rho|$ diverges, and higher derivatives grow faster. The required $\mathcal D$ seminorms therefore do not approach zero.

3. Fix a nonzero $\psi\in\mathcal D(\mathbb R)$ and set $\psi_n(x)=\psi(x-n)$. Why does $\psi_n$ fail to converge to zero in $\mathcal D(\mathbb R)$ even though every derivative converges pointwise to zero?

   Check

   The supports are translates $\operatorname{supp}\psi+n$ and escape every compact set. There is no single compact $K$ containing all supports, so the first condition for convergence in $\mathcal D$ fails.

References

• J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications, Chapters 2–3, 5, and 7, Birkhäuser, 2010. Book record. This is the structural source for test functions, distributions, convergence, and localization.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, MIT, 2022, Chapters 2–4. Open PDF. This is the teaching source for test-function convergence, continuity bounds, regular and Dirac distributions, localization, weak convergence, and support.
• A. S. Wightman, “How It Was Learned that Quantized Fields Are Operator-Valued Distributions,” Fortschritte der Physik 44 (1996), 143–178. Article record. This is the historical and QFT-facing source for interpreting fields through smeared operators rather than pointwise operator values.