Convergence of Greedy Algorithms in Banach Spaces

We study the convergence of greedy algorithms in Banach spaces. We construct an example of a smooth Banach space, where the $X$-greedy algorithm converges not for all dictionaries and initial vectors. We also study the $R$-greedy algorithm, which, along with the $X$-greedy algorithm, is a generalization of the simple greedy algorithm in Hilbert space. We prove its convergence for a certain class of Banach spaces. In particular, this class contains, the spaces $\ell^p$, $p\ge2$.