Editing Distribution (mathematics) (section)

====Topology on the space of distributions and its relation to the weak-* topology====

The set of all distributions on <math>U</math> is the [[continuous dual space]] of <math>C_c^\infty(U),</math> which when endowed with the [[Strong topology (polar topology)|strong dual topology]] is denoted by <math>\mathcal{D}'(U).</math> Importantly, unless indicated otherwise, the topology on <math>\mathcal{D}'(U)</math> is the [[strong dual topology]]; if the topology is instead the [[weak-* topology]] then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes <math>\mathcal{D}'(U)</math> into a [[Complete topological vector space|complete]] [[nuclear space]], to name just a few of its desirable properties.

Neither <math>C_c^\infty(U)</math> nor its strong dual <math>\mathcal{D}'(U)</math> is a [[sequential space]] and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is {{em|not}} enough to fully/correctly define their topologies).
However, a {{em|sequence}} in <math>\mathcal{D}'(U)</math> converges in the strong dual topology if and only if it converges in the [[weak-* topology]] (this leads many authors to use pointwise convergence to {{em|define}} the convergence of a sequence of distributions; this is fine for sequences but this is {{em|not}} guaranteed to extend to the convergence of [[Net (mathematics)|nets]] of distributions because a net may converge pointwise but fail to converge in the strong dual topology).
More information about the topology that <math>\mathcal{D}'(U)</math> is endowed with can be found in the article on [[spaces of test functions and distributions]] and the articles on [[Polar topology|polar topologies]] and [[dual system]]s.

A [[Linear map|{{em|linear}} map]] from <math>\mathcal{D}'(U)</math> into another [[locally convex topological vector space]] (such as any [[normed space]]) is [[Continuous function (topology)|continuous]] if and only if it is [[sequentially continuous]] at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general [[topological space]]s (for example, that are not also locally convex [[topological vector space]]s). The same is true of maps from <math>C_c^\infty(U)</math> (more generally, this is true of maps from any locally convex [[bornological space]]).