  We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space.
The aim is to classify subsets of Hilbert spaces that are contained in a
connected set of finite Hausdorff length. We do so by extending and improving
results of Peter Jones and Kate Okikiolu for sets in $\R^d$. Their results
formed the basis of quantitative rectifiability in $\R^d$. We prove a
quantitative version of the following statement: a connected set of finite
Hausdorff length (or a subset of one), is characterized by the fact that inside
balls at most scales around most points of the set, the set lies close to a
straight line segment (which depends on the ball). This is done via a quantity,
similar to the one introduced in \cite{Jones90}, which is a geometric analog of
the Square function. This allows us to conclude that for a given set $K$, the
$\ell_2$ norm of this quantity (which is a function of $K$) has size comparable
to a shortest (Hausdorff length) connected set containing $K$. In particular,
our results imply that, with a correct reformulation of the theorems, the
estimates in \cite{Jones90,Okik92} are {\bf independent of the ambient
dimension}.
