> [!definition] > > Let $U \subset \real^d$ be an [[Open Set|open set]]. A **distribution** $T$ on $U$ is a continuous linear functional on $\cd(U)$. The space $\cd'(U)$ is the [[Topological Dual|dual]] of the [[Space of Test Functions|space of test functions]], equipped with the [[Weak Topology|weak-* topology]]. > [!definition] > > Let $f \in \loci$ be a [[Locally Integrable|locally integrable]] function, then the mapping > $ > T_f: \cd(U) \to \complex \quad \phi \mapsto \angles{f, \phi} = \int_U f \phi dx > $ > is a distribution. If $T \in \cd'(U)$ and there exists $f \in \loci$ such that $T = T_f$, then the distribution $T$ is the **function** $f$. If there exists another function $g \in \loci$ such that $T_f = T_g$, then $f = g$ [[Almost Everywhere|almost everywhere]]. > > *Proof*. Let $V \subset U$ be a bounded [[Open Set|open set]], and let $\seq{\phi_n}$ be a sequence of [[Space of Test Functions|test functions]] such that $0 \le {\phi_n} \le \chi_{U}$ but $\phi_n \upto \int \chi_U$ pointwise. By the [[Dominated Convergence Theorem]], > $ > \angles{f, \chi_U} = \limv{n}\angles{f, \phi_n} = \limv{n}\angles{g, \phi_n} = \angles{g, \chi_U} > $ > Hence $E \mapsto \int_E (f - g)$ (when restricted to bounded sets) is a [[Signed Measure|signed measure]] that assigns $0$ to all bounded open sets. By the [[Lebesgue Differentiation Theorem]], $(f - g) = 0$ almost everywhere. # Operations on Distributions > [!definition] > > Let $z \in \real^d$, $f \in \loci(U + z)$ and $\phi \in \cd(U)$. If $\tau_z$ is the translation map by $z \in \real^d$, then > $ > \begin{align*} > \angles{\tau_zf, \phi} &= \int_{U} f(x - z)\phi(x)dx \\ > &= \int_{U + z} f(x )\phi(x + z)dx = \angles{f, \tau_{-z}\phi} > \end{align*} > $ > [[Linear Operator on Distribution|Extend this]] to distributions and define $\tau_z: \cd'(U + z) \to \cd'(U)$ by > $ > \angles{\tau_zT, \phi} = \angles{T, \tau_{-z}\phi} > $ > as the translation map on distributions. > [!definition] > > Let $S \in \laut{\real^d}$ be an invertible linear map, and $V = S^{-1}(U)$. Define $T: \loci{(V)} \to \loci{(U)}$ be defined by $f \mapsto f \circ T$, then for any $\phi \in \cd(U)$, > $ > \begin{align*} > \angles{Tf, \phi} &= \int_{U}(f \circ S)\phi = |\det S^{-1}|\int_V f (\phi \circ S^{-1}) \\ > &= |\det S^{-1}|\angles{f,\phi \circ S^{-1}} > \end{align*} > $ > Therefore for any $F \in \cd'(U)$, define the composition $F \circ S \in \cd'(V)$ by > $ > \angles{F \circ S, \phi} = |\det S^{-1}|\angles{f,\phi \circ S^{-1}} > $ > > In addition, define the reflection of a distribution about the origin by > $ > \anglesn{\td F, \phi} = \anglesn{F, \td \phi} > $ # Approximations > [!theorem] > > Let $T \in \cd'(U)$, then there exists [[Compactly Supported Distribution|compactly supported distributions]] $\seq{T_n} \subset \ce'(U)$ such that $T_n \to T$ in $\cd'(U)$. > > In particular, since compactly supported distributions can be approximated by $C_c^\infty$ functions, all distributions can be approximated by $C_c^\infty$ functions. > > *Proof*. Let $\seq{U_n}$ be an exhaustion of $U$ by precompact open sets. For each $n \in \nat$, let $\psi_n \in C_c^\infty(U)$ such that $\psi_n|_{\ol{U_n}} = 1$. Let $T_n = \psi_n T$, then for any $\phi \in \cd(U)$, there exists $N \in \nat$ such that $\supp{\phi} \subset U_N$ for all $n \ge N$. Therefore > $ > \angles{T_n, \phi} = \angles{T, \psi_n \phi} = \angles{T, \phi} > $ > for all $n \ge N$, and $T_n \to T$ in $\cd'(U)$. [^1]: Crude estimation for being in the same cube.