  Let $X_0$ be an infinite genus hyperbolic surface (whose boundary components,
if any, are closed geodesics or punctures) which has an upper bounded pants
decomposition. The length spectrum Teichm\"uller space $T_{ls}(X_0)$ consists
of all surfaces $X$ homeomorphic to $X_0$ such that the ratios of the
corresponding simple closed geodesics are uniformly bounded from below and from
above. Alessandrini, Liu, Papadopoulos and Su described the Fenchel-Nielsen
coordinates for $T_{ls}(X_0)$ and using these coordinates they proved that
$T_{ls}(X_0)$ is path connected. We use the Fenchel-Nielsen coordinates for
$T_{ls}(X_0)$ to induce a locally biLipschitz homeomorphism between
$l^{\infty}$ and $T_{ls}(X_0)$ (which extends analogous results by Fletcher and
by Allessandrini, Liu, Papadopoulos, Su and Sun for the unreduced and the
reduced $T_{qc}(X_0)$). Consequently, $T_{ls}(X_0)$ is contractible. We also
characterize the closure in the length spectrum metric of the quasiconformal
Teichm\"uller space $T_{qc}(X_0)$ in $T_{ls}(X_0)$.
