Conditional quasi-greedy bases in Hilbert and Banach spaces Gustavo GarrigosPrzemyslaw Wojtaszczyk 41A6541A4646B15thresholding greedy algorithmquasi-greedy basisconditional basis For quasi-greedy bases $\mathscr{B}$ in Hilbert spaces, we give---answering a question by Temlyakov---an improved bound of the associated conditionality constants $k_N(\mathscr{B})=O(\log N)^{1-\epsilon}$, for some $\epsilon>0$. We show the optimality of this bound with an explicit construction, based on a refinement of the method of Olevskii. This construction leads to other examples of quasi-greedy bases with large $k_N$ in Banach spaces, which are of independent interest. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5269 10.1512/iumj.2014.63.5269 en Indiana Univ. Math. J. 63 (2014) 1017 - 1036 state-of-the-art mathematics http://iumj.org/access/