In functional analysis and related areas of mathematics, a Smith space is a complete compactly generated locally convex topological vector space having a universal compact set, i.e. a compact set which absorbs every other compact set (i.e. for some ).
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 its stereotype dual space[4] is a Smith space,
- and vice versa, for any Smith space its stereotype dual space is a Banach space.
Smith spaces are special cases of Brauner spaces.
Examples
- As follows from the duality theorems, for any Banach space its stereotype dual space is a Smith space. The polar of the unit ball in is the universal compact set in . If denotes the normed dual space for , and the space endowed with the -weak topology, then the topology of lies between the topology of and the topology of , so there are natural (linear continuous) bijections
- If is infinite-dimensional, then no two of these topologies coincide. At the same time, for infinite dimensional the space is not barreled (and even is not a Mackey space if is reflexive as a Banach space[5]).
- If is a convex balanced compact set in a locally convex space , then its linear span possesses a unique structure of a Smith space with as the universal compact set (and with the same topology on ).[6]
- If is a (Hausdorff) compact topological space, and the Banach space of continuous functions on (with the usual sup-norm), then the stereotype dual space (of Radon measures on with the topology of uniform convergence on compact sets in ) is a Smith space. In the special case when is endowed with a structure of a topological group the space becomes a natural example of a stereotype group algebra.[7]
- A Banach space is a Smith space if and only if is finite-dimensional.
See also
Notes
- ^ Smith 1952.
- ^ Akbarov 2003, p. 220.
- ^ Akbarov 2009, p. 467.
- ^ The stereotype dual space to a locally convex space is the space of all linear continuous functionals endowed with the topology of uniform convergence on totally bounded sets in .
- ^ Akbarov 2003, p. 221, Example 4.8.
- ^ Akbarov 2009, p. 468.
- ^ 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.