Classical analysis is built on [functions](/page/Function): measurable maps from $\mathbb{R}^n$ (or an open subset) to $\mathbb{R}$. But many natural objects — the Dirac delta, derivatives of discontinuous functions, Green's functions of elliptic operators — are not functions in any reasonable sense. The theory of distributions, introduced by Laurent Schwartz in the 1940s and 1950s, resolves these difficulties by replacing pointwise evaluation with *continuous linear functionals on a space of [test functions](/page/Test%20Function)*. Instead of asking "what is the value of $u$ at $x$?", one asks "what is $T(\varphi)$ for every smooth, compactly supported $\varphi$?" This shift allows distributions to encompass delta masses, derivatives of discontinuous functions, and solutions to PDEs in the weakest possible sense.

## Motivation

[motivation] ### Why Functions Are Not Enough Consider the one-dimensional [wave equation](/page/Wave%20Equation) $u_{tt} - u_{xx} = 0$ on $\mathbb{R} \times (0, \infty)$ with initial data $u(x, 0) = f(x)$ and $u_t(x, 0) = 0$. D'Alembert's formula gives $u(x,t) = \tfrac{1}{2}(f(x+t) + f(x-t))$. If $f \in C^2(\mathbb{R})$, this is a classical solution: $u_{tt}$ and $u_{xx}$ exist pointwise and are equal. But if $f$ is merely continuous — say, a triangular pulse with a corner — then $u$ is continuous but not $C^2$, and the equation $u_{tt} = u_{xx}$ has no pointwise meaning at the corner. Yet the formula still describes the physically correct propagation of the wave. We need a framework in which $u$ "solves" the wave equation without requiring pointwise derivatives to exist. The situation is worse for nonlinear conservation laws. The inviscid Burgers equation $u_t + uu_x = 0$ with smooth initial data can develop discontinuities (shock waves) in finite time. After the shock forms, no classical solution exists, but the physics continues — the shock propagates according to the Rankine–Hugoniot conditions. The equation must be interpreted in a sense that allows discontinuous solutions and their "derivatives" to make sense. ### The Integration-by-Parts Idea The key observation is that *testing against smooth functions* can replace pointwise evaluation. Suppose $f \in C^{|\alpha|}(\Omega)$ and $\varphi \in C_c^\infty(\Omega)$. [Integration by parts](/theorems/210) gives \begin{align*} \int_\Omega (\partial^\alpha f)(x)\, \varphi(x) \, d\mathcal{L}^n(x) &= (-1)^{|\alpha|} \int_\Omega f(x)\, (\partial^\alpha \varphi)(x) \, d\mathcal{L}^n(x), \end{align*} with no [boundary](/page/Boundary) terms because $\varphi$ has compact support in $\Omega$. The right-hand side makes sense even when $f$ is merely locally integrable — one never differentiates $f$, only the smooth test function $\varphi$. This suggests defining the "derivative" of $f$ as the *rule* that assigns to each $\varphi$ the number $(-1)^{|\alpha|} \int f \, \partial^\alpha \varphi \, d\mathcal{L}^n$. More generally, any linear functional on test functions that is continuous in a suitable sense can serve as a "generalised function" — a distribution. ### From Weak Derivatives to Distributions The [weak derivative](/page/Weak%20Derivative), central to [Sobolev space](/page/Sobolev%20Spaces) theory, is a special case: $v \in L^p(\Omega)$ is the weak derivative $\partial^\alpha f$ if $\int v \varphi \, d\mathcal{L}^n = (-1)^{|\alpha|} \int f \, \partial^\alpha \varphi \, d\mathcal{L}^n$ for all $\varphi \in C_c^\infty(\Omega)$. But weak derivatives are required to be *functions* in $L^p$. Distributions remove this restriction: the distributional derivative of $T_f$ is the functional $\varphi \mapsto (-1)^{|\alpha|} \int f \, \partial^\alpha \varphi \, d\mathcal{L}^n$, which need not be representable by any locally integrable function. The Heaviside function $H$ has no weak derivative in any $L^p$ (its distributional derivative is the Dirac delta, which is not a function), but $T_H$ has a perfectly well-defined distributional derivative. The theory of distributions is the completion of the weak derivative idea: every locally integrable function generates a distribution whose distributional derivatives of all orders exist, regardless of the regularity of the original function. [/motivation]

## Test Functions

The definition of a distribution requires a space of "probe" functions against which distributions act. Smoothness ensures that integration by parts produces no error terms from non-differentiability; compact support ensures that boundary contributions vanish and all pairings are finite.

[definition: Space Of Test Functions] Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty open set. The **space of test functions** on $\Omega$ is \begin{align*} \mathcal{D}(\Omega) &:= \{\varphi \in C^\infty(\Omega) \mid \mathrm{supp}(\varphi) \subset \Omega \text{ is compact}\}. \end{align*} It carries the **[strict inductive limit topology](/page/Strict%20Inductive%20Limit%20Topology)**: the finest locally convex topology making each inclusion $\mathcal{D}_K(\Omega) \hookrightarrow \mathcal{D}(\Omega)$ continuous, where $\mathcal{D}_K(\Omega)$ is the [Fréchet space](/page/Fr%C3%A9chet%20Space) of smooth functions supported in a compact set $K \subset \Omega$. [/definition]

This topology is Hausdorff and complete but not metrizable. Despite the non-metrizability, a sequence $\varphi_k \to \varphi$ in $\mathcal{D}(\Omega)$ if and only if all supports lie in a single compact set and all derivatives converge uniformly ([Theorem 448](/theorems/448)). The full construction — bump functions, mollifiers, partitions of unity, and density results — is developed on the [Test Functions](/pages/1090) page.

## The Space of Distributions

With the test function space and its topology in hand, a distribution is defined as a continuous linear functional — a [linear map](/page/Linear%20Map) from $\mathcal{D}(\Omega)$ to the scalars that is continuous with respect to the strict inductive limit topology.

[definition: Distribution] Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty open set and equip $\mathcal{D}(\Omega)$ with the strict inductive limit topology. A **distribution** on $\Omega$ is a continuous linear map $T: \mathcal{D}(\Omega) \to \mathbb{R}$. [/definition]

The definition is [topological](/page/Topology): $T$ must send [open sets](/page/Open%20Set) in $\mathcal{D}(\Omega)$ to open sets in $\mathbb{R}$. Since the strict inductive limit topology on $\mathcal{D}(\Omega)$ is abstract and non-metrisable, it is natural to ask whether there are more concrete ways to verify that a given linear functional is a distribution. The following result provides two equivalent characterisations.

[quotetheorem:449]

The equivalence (1) $\Leftrightarrow$ (2) is the reason one can work with [sequences](/page/Sequence) throughout distribution theory despite the non-metrisability of $\mathcal{D}(\Omega)$: a linear functional is continuous if and only if it is sequentially continuous. This equivalence is a consequence of the LF-space structure (strict inductive [limit](/page/Limit) of Fréchet spaces) and would fail for general locally convex spaces. The equivalence (1) $\Leftrightarrow$ (3) gives a quantitative bound: the integer $N_K$ measures how many derivatives of the test function the distribution "uses" on the compact set $K$. When a single $N$ works for all compact [sets](/page/Set), $T$ is said to have **finite order** (at most $N$). For instance, regular distributions generated by $L^1_\mathrm{loc}$ functions have order $0$ (only the supremum of $\varphi$ appears), while the Dirac delta also has order $0$, and derivatives of the delta have order equal to the number of derivatives taken.

[definition: Space Of Distributions] Let $\Omega \subseteq \mathbb{R}^n$ be a non-empty open set. The **space of distributions** on $\Omega$, denoted $\mathcal{D}'(\Omega)$, is the set of all distributions on $\Omega$. It is a vector space under pointwise operations: \begin{align*} (T_1 + T_2)(\varphi) &:= T_1(\varphi) + T_2(\varphi), \\ (\lambda T)(\varphi) &:= \lambda \, T(\varphi) \end{align*} for $T_1, T_2, T \in \mathcal{D}'(\Omega)$, $\lambda \in \mathbb{R}$, and $\varphi \in \mathcal{D}(\Omega)$. [/definition]

The space $\mathcal{D}'(\Omega)$ is the continuous dual of the locally convex space $\mathcal{D}(\Omega)$, and as such it carries a natural topology: the [weak\* topology](/page/Weak*%20Topology) $\sigma(\mathcal{D}'(\Omega), \mathcal{D}(\Omega))$, which is the coarsest topology making every evaluation map $\operatorname{ev}_\varphi: T \mapsto T(\varphi)$ continuous. By the [Pointwise Characterisation of Weak Star Convergence](/theorems/504), a net (or sequence) $\{T_k\}$ converges to $T$ in $\sigma(\mathcal{D}'(\Omega), \mathcal{D}(\Omega))$ if and only if

\begin{align*} T_k(\varphi) \to T(\varphi) \quad \text{for every } \varphi \in \mathcal{D}(\Omega). \end{align*}

This is convergence "tested pointwise against all test functions," and it is the standard notion of convergence in distribution theory.

### Regular and Singular Distributions

Every locally integrable function gives rise to a distribution by integration, but not every distribution arises this way. The distinction between these two cases is fundamental.

[definition: Regular Distribution] Let $f \in L^1_\mathrm{loc}(\Omega)$. The **[regular distribution](/page/Regular%20Distribution)** generated by $f$ is the distribution $T_f \in \mathcal{D}'(\Omega)$ defined by \begin{align*} T_f: \mathcal{D}(\Omega) &\to \mathbb{R} \\ \varphi &\mapsto \int_\Omega f(x)\, \varphi(x) \, d\mathcal{L}^n(x). \end{align*} [/definition]