  We are interested in quantitative rectifiability results for subsets of
infinite dimensional Hilbert space $H$. We prove a version of Azzam and Schul's
$d$-dimensional Analyst's Travelling Salesman Theorem in this setting by
showing for any lower $d$-regular set $E \subseteq H$ that \[ \text{diam}(E)^d
+ \beta^d(E) \sim \mathscr{H}^d(E) + \text{Error}, \] where $\beta^d(E)$ give a
measure of the curvature of $E$ and the error term is related to the theory of
uniform rectifiability (a quantitative version of rectifiability introduced by
David and Semmes). To do this, we show how to modify the Reifenberg
Parametrization Theorem of David and Toro so that it holds in Hilbert space. As
a corollary, we show that a set $E \subseteq H$ is uniformly rectifiable if and
only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that
$E$ is bi-laterally well approximated by planes at most scales and locations.
