  Given a Banach space X and one of its compact sets F, we consider the problem
of finding a good n dimensional space X_n \subset X which can be used to
approximate the elements of F. The best possible error we can achieve for such
an approximation is given by the Kolmogorov width d_n(F)_X. However, finding
the space which gives this performance is typically numerically intractable.
Recently, a new greedy strategy for obtaining good spaces was given in the
context of the reduced basis method for solving a parametric family of PDEs.
The performance of this greedy algorithm was initially analyzed in A. Buffa, Y.
Maday, A.T. Patera, C. Prud'homme, and G. Turinici, "A Priori convergence of
the greedy algorithm for the parameterized reduced basis", M2AN Math. Model.
Numer. Anal., 46(2012), 595-603 in the case X = H is a Hilbert space. The
results there were significantly improved on in P. Binev, A. Cohen, W. Dahmen,
R. DeVore, G. Petrova, and P. Wojtaszczyk, "Convergence rates for greedy
algorithms in reduced bases Methods", SIAM J. Math. Anal., 43 (2011),
1457-1472. The purpose of the present paper is to give a new analysis of the
performance of such greedy algorithms. Our analysis not only gives improved
results for the Hilbert space case but can also be applied to the same greedy
procedure in general Banach spaces.
