A linear map \(u\colon \mathcal{D}(\Omega)\to \mathbb{R}\) is called distribution if for all compact \(K\subset \Omega\) there is a constant \(C=C(K)>0\) and \(m=m(K)\in \mathbb{N}\) such that for all test functions \(\varphi\in \mathcal{D}(\Omega)\) with \(\supp \varphi\subset K\) we have
\begin{equation*} \lvert u(\varphi)\rvert\le C\lVert \varphi \rVert_{C^m(\Omega)}. \end{equation*}We denote the space of distributions with \(\mathcal{D}'(\Omega)\).
Remarks
- \(\mathcal{D}'(\Omega)\) is like a dual space of \(\mathcal{D}(\Omega)\). However, \(\mathcal{D}(\Omega)\) is not a Banach space . The second criterion is a replacement for the continuity condition.