Smith space

In functional analysis and related areas of mathematics, Smith space is a complete compactly generated locally convex space $X$ having a compact set $K$ which absorbs every other compact set $T\subseteq X$ (i.e. $T\subseteq\lambda\cdot K$ for some $\lambda>0$).

Smith spaces are named after M. F. Smith, who introduced them as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces:

• for any Banach space $X$ its stereotype dual space $X^\star$ is a Smith space,

• and vice versa, for any Smith space $X$ its stereotype dual space $X^\star$ is a Banach space.