  The Analyst's Traveling Salesman Problem is to find a characterization of
subsets of rectifiable curves in a metric space. This problem was introduced
and solved in the plane by Jones in 1990 and subsequently solved in
higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the
infinite-dimensional Hilbert space $\ell_2$ by Schul in 2007. In this paper, we
establish sharp extensions of Schul's necessary and sufficient conditions for a
bounded set $E\subset\ell_p$ to be contained in a rectifiable curve from $p=2$
to $1<p<\infty$. While the necessary and sufficient conditions coincide when
$p=2$, we demonstrate that there is a strict gap between the necessary
condition and sufficient condition when $p\neq 2$. We also identify and correct
technical errors in the proof by Schul. This investigation is partly motivated
by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in
Banach spaces and complements work of Hahlomaa and recent work of David and
Schul on the Analyst's TSP in general metric spaces.
