  We present convergence estimates of two types of greedy algorithms in terms
of the metric entropy of underlying compact sets. In the first part, we measure
the error of a standard greedy reduced basis method for parametric PDEs by the
metric entropy of the solution manifold in Banach spaces. This contrasts with
the classical analysis based on the Kolmogorov n-widths and enables us to
obtain direct comparisons between the greedy algorithm error and the entropy
numbers, where the multiplicative constants are explicit and simple. The
entropy-based convergence estimate is sharp and improves upon the classical
width-based analysis of reduced basis methods for elliptic model problems. In
the second part, we derive a novel and simple convergence analysis of the
classical orthogonal greedy algorithm for nonlinear dictionary approximation
using the metric entropy of the symmetric convex hull of the dictionary. This
also improves upon existing results by giving a direct comparison between the
algorithm error and the metric entropy.
