Smith space

In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space $X$ having a universal compact set, i.e. 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 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 $X$ its stereotype dual space^[4] $X^{\star }$ is a Smith space,

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

Smith spaces are special cases of Brauner spaces.

Examples

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

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

If

X

is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional

X

the space

X^{\star }

is not barreled (and even is not a Mackey space if

X

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

[1]
Smith 1952.
[2]
Akbarov 2003, p. 220.
[3]
Akbarov 2009, p. 467.
[4]
The stereotype dual space to a locally convex space $X$ is the space $X^{\star }$ of all linear continuous functionals $f:X\to \mathbb {C}$ endowed with the topology of uniform convergence on totally bounded sets in $X$ .
[5]
Akbarov 2003, p. 221, Example 4.8.
[6]
Akbarov 2009, p. 468.
[7]
Akbarov 2003, p. 272.