nLab liquid vector space

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

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<p10\lt p\le 1 (i.e., the non-convex case) into the picture.

Definition

A condensed abelian group VV is pp-liquid (0<p10\lt p\le 1) if for every compact Hausdorff topological space SS and every morphism of condensed sets f:SVf\colon S\to V there is a unique morphism of condensed abelian groups M <p(S)VM_{\lt p}(S)\to V that extends ff along the inclusion SM <p(S)S\to M_{\lt p}(S).

Here for a compact Hausdorff topological space SS and for any pp such that 0<p10\lt p\le 1 we have

M <p(S)= q<pM q(S),M_{\lt p}(S)=\bigcup_{q\lt p}M_q(S),

where

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

where

M(S) pC=lim iM(S i) pC,M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},

where S iS_i are finite sets such that

S=lim iS iS = \lim_i S_i

and

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

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

References

See at condensed mathematics.

Last revised on May 28, 2022 at 11:17:02. See the history of this page for a list of all contributions to it.