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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 27th 2022

Created:

## 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 doesnâ€™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_{ that extends $f$ along the inclusion $S\to M_{.

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

$M_{

where

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

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 l^p-norm at most $C$.

## References

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 28th 2022
• (edited May 28th 2022)

have fixed the link to $\ell^p$-norm:

  [[p-norm|$\ell^p$-norm]]