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It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is the subgroup $L^2(I,\mathbb{Z})$ of the real Hilbert space of all $L^2$ real valued functions on the unit interval $I:=[0,1].$

Indeed, any element $\phi$ of $L^2(I,\mathbb{Z})$ is connected to the origin by the path $\gamma:I\ni t\mapsto \phi\chi_{[0,t]}\in L^2(I,\mathbb{Z}),$ where $\chi_{[0,t]}$ is the characteristic function of the interval $[0,t].$ Actually, up to a reparametrization, this path is also $1/2$-Hölder continuous. (Indeed, if $\sigma:[0,1]\to\big[0,\ 1+ \|\phi\| _2^2\ \big]$ is the strictly increasing, surjective continuous map $t\mapsto t+\int_0^t \phi^2dx$, then $\|\gamma(t)-\gamma(t')\|_2\le|\sigma(t)-\sigma(t')|^{1/2}$, meaning that $\gamma\circ\sigma^{-1}$ is $1/2$-Hölder continuous).
So we may say that $L^2(I,\mathbb{Z})$ is even $1/2$-Hölder-path-connected, though it is certainly not a linear subspace.

It is also not hard to see that the Hölder exponent $1/2$ is critic: any closed subgroup $G$ of a Hilbert space $H$, which is connected by $\alpha$-Hölder paths, with $\alpha > 1/2,$ is necessarily a linear space. (Reason: as a consequence of the generalized parallelogram identity, it turns out that the lattice generated by $n$ vectors $g_1,\dots,g_n$ in $H$, with norms $\|g_k\|\leq r,$ is a $rn^{1/2}$-net in their linear span. In particular, if $\gamma:[0,1]\to G$ is an $\alpha$-Hölder path, for any $n\in\mathbb{N},$ the $n$ elements $g_{k,n}:=\gamma(\frac{k+1}{n})-\gamma(\frac{k}{n})\in G,\quad k=0,\dots,n-1$ are a $Cn^{\alpha-1/2}$-net Cn^{1/2 - \alpha }$-net in their linear span. Since$G$is closed this implies that it is a cone, hence a linear subspace). I find this quite nice, but at this point some questions arise quite naturally. Let$H$be the infinite dimensional real separable Hilbert space. • Let$0 < \alpha < 1/2.$Are there closed additive subgroups of$H$which are connected by$\alpha$-Hölder paths, but not by$\beta$-Hölder paths for any$\beta >\alpha \ $? • More generally: connected / non-connected w.r.to paths with given modulus of continuity? Are there closed, connected, not path-connected additive subgroups? • Are these objects just pathologies/curiosities of the mathematical Zoo, or did anybody gave an application of them to functional analysis? 3 added 292 characters in body; added 2 characters in body It is a classical result that a closed and connected additive subgroup of$\mathbb{R}^n$is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is the subgroup$L^2(I,\mathbb{Z})$of the real Hilbert space of all$L^2$real valued functions on the unit interval$I:=[0,1].$Indeed, any element$\phi$of$L^2(I,\mathbb{Z})$is connected to the origin by the path$I\ni \gamma:I\ni t\mapsto \phi\chi_{[0,t]}\in L^2(I,\mathbb{Z}),$where$\chi_{[0,t]}$is the characteristic function of the interval$[0,t].$Actually, up to a reparametrization, this path is also$1/2$-Hölder continuous. (Indeed, if$\sigma:[0,1]\to\big[0,\ 1+ \|\phi\| _2^2\ \big]$is the strictly increasing, so surjective continuous map$t\mapsto t+\int_0^t \phi^2dx$, then$\|\gamma(t)-\gamma(t')\|_2\le|\sigma(t)-\sigma(t')|^{1/2}$, meaning that$\gamma\circ\sigma^{-1}$is$1/2$-Hölder continuous). So we may say that$L^2(I,\mathbb{Z})$is even$1/2$-Hölder-path-connected, though it is certainly not a linear subspace. It is also not hard to see that the Hölder exponent$1/2$is critic: any closed subgroup$G$of a Hilbert space$H$, which is connected by$\alpha$-Hölder paths, with$\alpha > 1/2,$is necessarily a linear space. (Reason: as a consequence of the generalized parallelogram identity, it turns out that the lattice generated by$n$vectors$g_1,\dots,g_n$in$H$, with norms$\|g_k\|\leq r,$is a$rn^{1/2}$-net in their linear span. In particular, if$\gamma:[0,1]\to G$is an$\alpha$-Hölder path, for any$n\in\mathbb{N},$the$n$elements$g_{k,n}:=\gamma(\frac{k+1}{n})-\gamma(\frac{k}{n})\in G,\quad k=0,\dots,n-1$are a$Cn^{\alpha-1/2}$-net in their linear span. Since$G$is closed this implies that it is a cone, hence a linear subspace). I find this quite nice, but at this point some questions arise quite naturally. Let$H$be the infinite dimensional real separable Hilbert space. • Let$0 < \alpha < 1/2.$Are there closed additive subgroups of$H$which are connected by$\alpha$-Hölder paths, but not by$\beta$-Hölder paths for any$\beta >\alpha \ $? • More generally: connected / non-connected w.r.to paths with given modulus of continuity? Are there closed, connected, not path-connected additive subgroups? • Are these objects just pathologies/curiosities of the mathematical Zoo, or did anybody gave an application of them to functional analysis? 2 added 12 characters in body It is a classical result that a closed and connected additive subgroup of$\mathbb{R}^n$is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is the subgroup$L^2(I,\mathbb{Z})$of the real Hilbert space of all$L^2$real valued functions on the unit interval$I:=[0,1].$Indeed, any element$\phi$of$L^2(I,\mathbb{Z})$is connected to the origin by the path$I\ni t\mapsto \phi\chi_{[0,t]}\in L^2(I,\mathbb{Z}),$where$\chi_{[0,t]}$is the characteristic function of the interval$[0,t].$Actually, up to a reparametrization, this path is also$1/2$-Hölder continuous, so we may say that$L^2(I,\mathbb{Z})$is even$1/2$-Hölder-path-connected, though it is certainly not a linear subspace. It is also not hard to see that the Hölder exponent$1/2$is critic: any closed subgroup$G$of a Hilbert space$H$, which is connected by$\alpha$-Hölder paths, with$\alpha > 1/2,$is necessarily a linear space. (Reason: as a consequence of the generalized parallelogram identity, it turns out that the lattice generated by$n$vectors$g_1,\dots,g_n$in$H$, with norms$\|g_k\|\leq r,$is a$rn^{1/2}$-net in their linear span. In particular, if$\gamma:[0,1]\to G$is an$\alpha$-Hölder path, for any$n\in\mathbb{N},$the$n$elements$g_{k,n}:=\gamma(\frac{k+1}{n})-\gamma(\frac{k}{n})\in G,\quad k=0,\dots,n-1$are a$Cn^{\alpha-1/2}$-net in their linear span. Since$G$is closed this implies that it is a cone, hence a linear subspace). I find this quite nice, but at this point some questions arise quite naturally. Let$H$be the infinite dimensional real separable Hilbert space. • Let$0 < \alpha < 1/2.$Are there closed additive subgroups of$H$which are connected by$\alpha$-Hölder paths, but not by$\beta$-Hölder paths for any$\beta >\alpha \ \$?

• More generally: connected / non-connected w.r.to paths with given modulus of continuity? Are there closed, connected, not path-connected additive subgroups?

• Are these objects just pathologies/curiosities of the mathematical Zoo, or did anybody gave an application of them to functional analysis?

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