We analyze the hypercyclicity, chaoticity, and spectral structure of (bounded
and unbounded) weighted backward shifts in a nonclassical sequence space, which
the space $l_1$ of summable sequences is both isometrically isomorphic to and
continuously and densely embedded into.
Based on the weighted backward shifts, we further construct new bounded and
unbounded linear hypercyclic and chaotic operators both in the nonclassical
sequence space and the classical space $l_1$, including those that are
hypercyclic but not chaotic.