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 $Cn^{1/2 - \alpha }$-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?