  Given a metric space $X$, an Analyst's Traveling Salesman Theorem for $X$
gives a quantitative relationship between the length of a shortest curve
containing any subset $E\subseteq X$ and a multi-scale sum measuring the
``flatness'' of $E$. The first such theorem was proven by Jones for $X =
\mathbb{R}^2$ and extended to $X = \mathbb{R}^n$ by Okikiolu, while an
analogous theorem was proven for Hilbert space, $X = H$, by Schul. Bishop has
since shown that if one considers Jordan arcs, then the quantitative
relationship given by Jones' and Okikioulu's results can be sharpened. This
paper gives a full proof of Schul's original necessary half of the traveling
salesman theorem in Hilbert space and provides a sharpening of the theorem's
quantitative relationship when restricted to Jordan arcs analogous to Bishop's
aforementioned sharpening in $\mathbb{R}^n$.
