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We define a manifold $M$ where objects $c\in M$ are curves, which weparameterize 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 alldeformations $h:S^1\to R^n$ of $c$. In this paper we study geometries on themanifold 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 thecompletion of the space of smooth curves. As a bonus, we prove that theFr\'echet distance of curves (see arXiv:math.DG/0312384) coincides with thedistance induced by the ``Finsler $L^\infinity$ metric'' defined in \S2.2 inarXiv:math.DG/0412454

Properties of Sobolev-type metrics in the space of curves