nLab liquid vector space

Idea

An alternative to complete topological vector spaces in the framework of condensed mathematics.

Roughly, completeness is expressed as ability to integrate with respect to Radon measures.

This doesn’t quite work as stated, and to make this rigorous one has to bring L^p-spaces for $0\lt p\le 1$ (i.e., the non-convex case) into the picture.

Definition

A condensed abelian group $V$ is $p$-liquid ($0\lt p\le 1$) if for every compact Hausdorff topological space $S$ and every morphism of condensed sets $f\colon S\to V$ there is a unique morphism of condensed abelian groups $M_{\lt p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{\lt p}(S)$.

Here for a compact Hausdorff topological space $S$ and for any $p$ such that $0\lt p\le 1$ we have

$$M_{\lt p}(S)=\bigcup_{q\lt p}M_q(S),$$

where

$$M_p(S)=\bigcup_{C\gt 0}M(S)_{\ell^p\le C},$$

where

$$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$

where $S_i$ are finite sets such that

$$S = \lim_i S_i$$

and

$$M(F)_{\ell^p\le C}$$

for a finite set $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with $\ell^p$-norm at most $C$.

Related concepts

• solid module

• gaseous vector space

References

See at condensed mathematics.