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 M. F. Smith,^[1] 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:^[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.

Notes

1. M. F. Smith (1952).
2. S.S.Akbarov (2003).
3. S.S.Akbarov (2009).
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 $X$.

References

• Schaefer, Helmuth H. (1966). Topological vector spaces. New York: The MacMillan Company. ISBN 0-387-98726-6.

• Robertson, A.P.; Robertson, W.J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press.
• 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. doi:10.1007/s10958-009-9646-1. (subscription required (help)).