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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 27th 2022



    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.


    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 Unknown characterp(S)VM_{<p}(S)\to V that extends ff along the inclusion SM Unknown characterp(S)S\to M_{<p}(S).

    Here for a compact Hausdorff topological space SS and for any pp such that 0Unknown characterp10<p\le 1 we have

    M Unknown characterp(S)= qUnknown characterpM q(S),M_{<p}(S)=\bigcup_{q<p}M_q(S),


    M p(S)= CUnknown character0M(S) pC,M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},


    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


    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 l^p-norm at most CC.


    See condensed mathematics.

    v1, current

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

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


    diff, v3, current