Sobolev--type metrics in the space of curves

We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to R^n$ of $c$. In this paper we study geometries on the manifold of curves, provided by Sobolev--type metrics $H^j$. We study $H^j$ type metrics for the cases $j=1,2$; we prove estimates, and characterize the completion of the space of smooth curves. As a bonus, we prove that the Fr\'echet distance of curves (see arXiv:math.DG/0312384) coincides with the distance induced by the ``Finsler $L^\infinity$ metric'' defined in \S2.2 in arXiv:math.DG/0412454.