  The classical Beurling-Helson-Lowdenslager theorem characterizes the
shift-invariant subspaces of the Hardy space $H^{2}$ and of the Lebesgue space
$L^{2}$. In this paper, which is self-contained, we define a very general class
of norms $\alpha$ and define spaces $H^{\alpha}$ and $L^{\alpha}.$ We then
extend the Beurling-Helson-Lowdenslager invariant subspace theorem. The idea of
the proof is new and quite simple; most of the details involve extending basic
well-known $\left \Vert {}\right \Vert _{p}$-results for our more general
norms.
