The exhaustion metric on a function space over a plane domain

Fix a plane domain $\mathrm{\Omega }$ and a compact exhaustion $(K_n{)}_{n\ge 1}$ of $\mathrm{\Omega }$; for this page the canonical exhaustion of Every plane domain has the canonical nested compact exhaustion by distance and radius cutoffs is the default choice. For functions $f,g:\mathrm{\Omega }\to \mathbb{C}$, set $s_n(f,g):=\{\begin{array}{ll}0,& K_n=\mathrm{\varnothing },\\ {\sup }_{z\in K_n}\mathrm{\mid }f(z)-g(z)\mathrm{\mid },& K_n\mathrm{\ne }\mathrm{\varnothing }\mathrm{.}\end{array}$ and define the exhaustion metric by $d_K(f,g):={\sum }_{n\ge 1}{2}^{-n}\min (1,s_n(f,g))\mathrm{.}$

Here the supremum is taken in $[0,\mathrm{\infty }]$, so $s_n(f,g)=+\mathrm{\infty }$ is allowed, and $\min (1,+\mathrm{\infty }):=1$. Thus each summand is a real number in $[0,2^{-n}]$, so the series converges absolutely. The metric is used on spaces of continuous or holomorphic functions on $\mathrm{\Omega }$; the next theorem shows that its convergent sequences are exactly the locally uniformly convergent ones.