Schwartz Distributions

Abstract

In our approach Schwartz distributions are defined as continuous linear functionals on an appropriate test function space. Thus this chapter starts with a reminder of some basic facts about the topological dual of an Hausdorff locally convex topological vector space. Then we proceed to define (general) Schwartz distributions T as continuous linear functionals on $\mathcal{D}(\Omega)$, i.e., as an element of the topological dual $\mathcal{D}^{\prime}(\Omega)$ of $\mathcal{D}(\Omega)$. Similarly tempered distributions are defined as elements of the topological dual $\mathcal{S}^{\prime}(\Omega)$ of the space $\mathcal{S}(\Omega)$. Finally compactly supported distributions are elements of the topological dual $\mathcal{E}^{\prime}(\Omega)$ of $\mathcal{E}(\Omega)$. Furthermore regular distributions are introduced as the embedding of the space of locally integrable functions $L^1_{loc}(\Omega)$ on Ω into $\mathcal{D}^{\prime}(\Omega)$. Standard examples of distributions illustrate these definitions. When the distribution space $\mathcal{D}^{\prime}(\Omega)$ is equipped with the weak topology $\sigma=\sigma(\mathcal{D}^{\prime}(\Omega),\mathcal{D}(\Omega))$ it is sequentially complete. Localization of distributions is defined through the localized test functions in $\mathcal{D}(\Omega)$. The resulting concept of support of a distribution is compatible with the notion of support of functions in $L^1_{loc}(\Omega)$ through the standard embedding.