  We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary
complete seminormalized biorthogonal systems in Banach spaces. The bounds are
given only in terms of the upper democracy functions of the basis and its dual.
We also show that these estimates are equivalent to embeddings between the
given Banach space and certain discrete weighted Lorentz spaces. Finally, the
asymptotic optimality of these inequalities is illustrated in various examples
of non necessarily quasi-greedy bases.
