Infunctional analysis and related areas ofmathematics, aSmith space is acompletecompactly generatedlocally convex topological vector space${\displaystyle X}$ having auniversal 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 afterMarianne Ruth Freundlich Smith, who introduced them^[1] as duals toBanach spaces in some versions of duality theory fortopological vector spaces. All Smith spaces arestereotype and are in the stereotype duality relations withBanach 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 ofBrauner spaces.

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

${\displaystyle X^{\ast }\to X^{\star }\to X^{\prime }.}$

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 notbarreled (and even is not aMackey space if${\displaystyle X}$ isreflexive as a Banach space^[5]).

• If${\displaystyle K}$ is aconvexbalancedcompact set in alocally convex space${\displaystyle Y}$, then itslinear span${\displaystyle \mathbb{C}K=\mathrm{span}(K)}$ 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${\displaystyle \mathcal{C}(M)}$ theBanach space of continuous functions on${\displaystyle M}$ (with the usual sup-norm), then the stereotype dual space${\displaystyle {\mathcal{C}}^{\star }(M)}$ (ofRadon 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 atopological group the space${\displaystyle {\mathcal{C}}^{\star }(G)}$ becomes a natural example of astereotype group algebra.^[7]
• ABanach space${\displaystyle X}$ is a Smith space if and only if${\displaystyle X}$ is finite-dimensional.

• Stereotype space
• Brauner space

• 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.

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