On the relation between linear n-widths and approximation numbers

AbstractWe show that for a compact absolutely convex subset of a normed space the linear n-width obtained with the help of bounded linear rank n operators, the one obtained by arbitrary linear rank n operators, and the corresponing approximation number of the associated embedding, all coincide. This is achieved by a version of the principle of local reflexivity for spaces connected with embedding operators. We also give a counterexample showing that for relatively compact absolutely convex sets equality does not hold any longer and the discrepancy can be of the (maximally possible) order n12