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 Marianne Ruth Freundlich 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.

See also

• Stereotype space
• Brauner space

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.
• 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. (Subscription required (help)).
• Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PhD). Radboud University.