On left democracy function

Abstract

We continue the study undertaken in [1] of left democracy function $$h_l(N)=\inf_{\#\Lambda=N}||\sum_{n\in \Lambda_N}x_n||$$ of an unconditional basis in a Banach space $X$. We provide an example of a basis with $h_l$ non-doubling. Then we show that for bases with non-doubling $h_l$ the greedy projection is not optimal. Together with results from [1] improved by C. Cabrelli, G. Garrigós, E. Hernandez and U. Molter we get that the basis is greedy if and only if the greedy projection is optimal.