Closed connected additive subgroups of the Hilbert space. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:27:55Z http://mathoverflow.net/feeds/question/45322 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45322/closed-connected-additive-subgroups-of-the-hilbert-space Closed connected additive subgroups of the Hilbert space. Pietro Majer 2010-11-08T16:48:41Z 2011-10-18T10:15:36Z <p>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 $I:=[0,1].$ </p> <p>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).<br> 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.</p> <p>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).</p> <p>I find this quite nice, but at this point some questions arise quite naturally. Let $H$ be the infinite dimensional real separable Hilbert space.</p> <ul> <li><p>Let $0 &lt; \alpha &lt; 1/2.$ Are there closed additive subgroups of $H$ which are connected by $\alpha$-Hölder paths, but not by $\beta$-Hölder paths for any $\beta >\alpha \$? </p></li> <li><p>More generally: connected / non-connected w.r.to paths with given modulus of continuity? Are there closed, connected, not path-connected additive subgroups? </p></li> <li><p>Are these objects just pathologies/curiosities of the mathematical Zoo, or did anybody gave an application of them to functional analysis? </p></li> </ul>