Smith space

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.

See also

• Stereotype space
• Brauner space

Notes

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

References

• Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces". Annals of Mathematics. 56 (2): 248–253. doi:10.2307/1969798. JSTOR 1969798.
• Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. S2CID 115297067.
• Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. S2CID 115153766.
• Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PDF) (PhD). Radboud University.