  We prove some results on the rate of convergence of greedy algorithms, which
provide expansions. We consider both the case of Hilbert spaces and the more
general case of Banach spaces. The new ingredient of the paper is that we bound
the error of approximation by the product of both norms -- the norm of $f$ and
the $A_1$-norm of $f$. Typically, only the $A_1$-norm of $f$ is used. In
particular, we establish that some greedy algorithms (Pure Greedy Algorithm
(PGA) and its generalizations) are as good as the Orthogonal Greedy Algorithm
(OGA) in this new sense of the rate of convergence, while it is known that the
PGA is much worth than the OGA in the standard sense.
