Function space

A function space is a collection of functions with common domain and codomain, often supplied with a vector space structure and a norm or topology. Examples include L2 spaces and Sobolev spaces.

For a locally compact Hausdorff space $X$, $C_c(X)$ consists of continuous functions whose support is compact. A positive linear functional on this space, bounded on each fixed compact support, is represented by a Radon measure. For a locally compact group with Haar measure, these functions are dense in $L^p$ for $1\le p<\infty$.

The Skorokhod space consists of real paths that are right-continuous and have a finite left limit at every positive time. Its $J_1$ topology allows small changes of time: $x_n\to x$ when there are increasing homeomorphisms ${\lambda }_n$ of $[0,1]$ such that $\Vert {\lambda }_n-\mathrm{id}{\Vert }_{\infty }\to 0$ and $\Vert x_n\circ {\lambda }_n-x{\Vert }_{\infty }\to 0$, where $\mathrm{id}$ denotes the identity time map. This topology aligns nearby jumps rather than insisting on uniform closeness at identical times. The step-path version of the Donsker invariance principle converges here to continuous Brownian motion; its polygonal version uses the ordinary supremum norm on continuous paths.