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

Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them^[1] 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:^[2][3]

• for any Banach space ${\displaystyle X}$ its stereotype dual space^[4] ${\displaystyle X^{\star ))$ is a Smith space,

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

Smith spaces are special cases of Brauner spaces.

Examples

• As follows from the duality theorems, for any Banach space ${\displaystyle X}$ its stereotype dual space ${\displaystyle X^{\star ))$ is a Smith space. The polar ${\displaystyle K=B^{\circ ))$ of the unit ball ${\displaystyle B}$ in ${\displaystyle X}$ is the universal compact set in ${\displaystyle X^{\star ))$. If ${\displaystyle X^{*))$ denotes the normed dual space for ${\displaystyle X}$, and ${\displaystyle X'}$ the space ${\displaystyle X^{*))$ endowed with the ${\displaystyle X}$-weak topology, then the topology of ${\displaystyle X^{\star ))$ lies between the topology of ${\displaystyle X^{*))$ and the topology of ${\displaystyle X'}$, so there are natural (linear continuous) bijections

${\displaystyle X^{*}\to X^{\star }\to X'.}$

If ${\displaystyle X}$ is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional ${\displaystyle X}$ the space ${\displaystyle X^{\star ))$ is not barreled (and even is not a Mackey space if ${\displaystyle X}$ is reflexive as a Banach space^[5]).

• If ${\displaystyle K}$ is a convex balanced compact set in a locally convex space ${\displaystyle Y}$, then its linear span ${\displaystyle {\mathbb {C} }K=\operatorname {span} (K)}$ possesses a unique structure of a Smith space with ${\displaystyle K}$ as the universal compact set (and with the same topology on ${\displaystyle K}$).^[6]
• If ${\displaystyle M}$ is a (Hausdorff) compact topological space, and ${\displaystyle {\mathcal {C))(M)}$ the Banach space of continuous functions on ${\displaystyle M}$ (with the usual sup-norm), then the stereotype dual space ${\displaystyle {\mathcal {C))^{\star }(M)}$ (of Radon measures on ${\displaystyle M}$ with the topology of uniform convergence on compact sets in ${\displaystyle {\mathcal {C))(M)}$) is a Smith space. In the special case when ${\displaystyle M=G}$ is endowed with a structure of a topological group the space ${\displaystyle {\mathcal {C))^{\star }(G)}$ becomes a natural example of a stereotype group algebra.^[7]
• A Banach space ${\displaystyle X}$ is a Smith space if and only if ${\displaystyle X}$ is finite-dimensional.