Distributions extend functions by testing against smooth compactly supported probes. Derivatives are moved onto the test function, so nonsmooth objects can still be differentiated.
Test functions and distributions
A distribution is defined by how it acts on test functions. The topology on the test-function space is chosen so that all derivatives and local supports are visible.
Definition. For an open \(U\subset\mathbb R^d\), \(C_c^\infty(U)\) is the space of smooth compactly supported test functions. A distribution on \(U\) is a continuous linear functional \(T:C_c^\infty(U)\to\mathbb F\), written \(\langle T,\varphi\rangle\).
Example. If \(f\in L^1_{\mathrm{loc}}(U)\), then \(T_f(\varphi)=\int_U f(x)\varphi(x)\,\mathrm{d}x\) defines a distribution. Continuity follows because on each compact support \(K\), \(|T_f(\varphi)|\le\|f\|_{L^1(K)}\|\varphi\|_\infty\).
Distributional derivatives
Weak differentiation is obtained by moving derivatives from the object onto the test function. The sign convention is chosen so that smooth functions give the usual derivative.
Definition. For a multi-index \(\alpha\), define \(\partial^\alpha T\) by \(\langle\partial^\alpha T,\varphi\rangle=(-1)^{|\alpha|}\langle T,\partial^\alpha\varphi\rangle\).
Proposition. If \(f\in C^{|\alpha|}(U)\), then the distributional derivative of \(T_f\) equals \(T_{\partial^\alpha f}\).
Proof. Repeated integration by parts gives \(\int_U \partial^\alpha f\,\varphi\,\mathrm{d}x=(-1)^{|\alpha|}\int_U f\,\partial^\alpha\varphi\,\mathrm{d}x\), because \(\varphi\) has compact support in \(U\) and all boundary terms vanish. This is exactly the defining identity for \(\partial^\alpha T_f\).
Weak derivatives
Weak derivatives identify when a distributional derivative is represented by an ordinary \(L^1_{\mathrm{loc}}\) function. This is the entry point to Sobolev spaces.
Definition. For \(f,g\in L^1_{\mathrm{loc}}(U)\), \(g\) is the weak derivative \(\partial^\alpha f\) if \(\int_U g\varphi\,\mathrm{d}x=(-1)^{|\alpha|}\int_U f\,\partial^\alpha\varphi\,\mathrm{d}x\) for all \(\varphi\in C_c^\infty(U)\).
Proposition. Weak derivatives are unique up to equality a.e.
Proof. If \(g_1,g_2\) both represent the same weak derivative, then \(\int_U(g_1-g_2)\varphi\,\mathrm{d}x=0\) for all test functions. The fundamental lemma of distributions says a locally integrable function with zero action on all test functions is zero a.e.; it follows by testing against mollified approximations of indicators on compact subsets.
Convolution with test functions
Convolution with a smooth compactly supported function regularizes distributions. It is the basic device for approximating rough objects by smooth ones.
Proposition. If \(T\) is a distribution and \(\rho\in C_c^\infty(\mathbb R^d)\), then \(T*\rho\) is a smooth function and \(\partial^\alpha(T*\rho)=T*(\partial^\alpha\rho)\).
Proof. Define \((T*\rho)(x)=\langle T,\rho(x-\cdot)\rangle\). The map \(x\mapsto\rho(x-\cdot)\) is smooth as a map into the test-function topology on compact \(x\)-sets. Continuity and linearity of \(T\) allow differentiation under the distribution bracket, giving \(\partial^\alpha_x(T*\rho)(x)=\langle T,\partial^\alpha\rho(x-\cdot)\rangle\).
Delta distribution
The delta distribution is not a function but an evaluation functional. It is the simplest object showing why distributions genuinely enlarge the notion of function.
Example. The Dirac mass \(\delta_a\) is defined by \(\langle\delta_a,\varphi\rangle=\varphi(a)\). Its derivative satisfies \(\langle\partial_i\delta_a,\varphi\rangle=-\partial_i\varphi(a)\). This is not a function but is a continuous linear functional on the test-function space.
Distributional convergence
Distributional convergence is convergence against every test function. It is weak enough to capture many singular limits while still preserving differentiation.
Definition. A sequence \(T_n\) converges to \(T\) in distributions if \(\langle T_n,\varphi\rangle\to\langle T,\varphi\rangle\) for every \(\varphi\in C_c^\infty(U)\). This is weak-star convergence against test functions.