nLab Smith space (functional analysis)

For the generalized smooth space of the same name see at Smith space (generalized smooth space).

Idea

Definition

In the context of functional analysis, a Smith space is a complete locally convex topological $\mathbb{R}$-vector space $V$ that admits a compact absolutely convex subset $K \subset V$ such that $V = \bigcup_{c \gt 0} c K$ with the induced compactly generated topology on $V$.

Examples

• For a compact absolutely convex set, $S$, the space of measures, $M(S)$, with the compactly generated topology is a Smith space.

Properties

The category of Smith spaces is equivalent to the opposite of the category of Banach spaces. More precisely, if $V$ is a Banach space, then $Hom(V,\mathbb{R})$ is a Smith space; and if $W$ is a Smith space, then $Hom(W,\mathbb{R})$ is a Banach space, where in both cases we endow the dual space with the compact-open topology. The corresponding biduality maps are isomorphisms.

Smith spaces are more natural than Banach spaces from the perspective of condensed mathematics because they are controlled by a nice compact subset instead of a nice open subset (the unit ball) as in the Banach setting. Also, Banach spaces are filtered colimits of Smith spaces while Smith spaces are filtered limits of Banach spaces. Filtered colimits have better algebraic and homological properties.

Smith spaces embed fully faithfully in condensed R-vector spaces, but Smith spaces do not form an abelian category. There is an induced functor from Waelbrock’s enlarged (abelian) category of quotients of Smith spaces, but that functor is not fully faithful.

Smith spaces are the same as Waelbrock dual spaces.

A Banach space is a Smith space if and only if it is finite-dimensional.

References

The original treatment is in

• Marianne Smith, The Pontrjagin duality theorem in linear spaces, Annals of Mathematics. 56(2): 248–253, (doi:10.2307/1969798).

In the context of condensed mathematics, see

• Peter Scholze, Lectures on Analytic Geometry, (pdf)