Smith space
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 .
Smith spaces are named after
, 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:
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 bijections
- 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.
- If is a compact topological space, and the Banach space of continuous functions on, then the stereotype dual space 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.
- A Banach space is a Smith space if and only if is finite-dimensional.