Weak Convergence of a Greedy Algorithm and the WN-Property.

We study the weak convergence of a greedy algorithm of approximation by a given set in a Banach space. It is proved that the greedy algorithm of approximation by a strongly norm-reducing set in a uniformly smooth Banach space with the WN-property weakly converges. In an arbitrary separable Banach space without the WN-property, we construct an example of a strongly norm-reducing set such that the greedy algorithm of approximation by this set does not weakly converge for some initial element. [ABSTRACT FROM AUTHOR]