In Chapters 4 to 7 we considered approximation problems in which the approximating set (or the sequence of sets) is fixed. In particular, exact results for estimating the approximation error in functional classes by elements of finite dimensional polynomial or spline subspaces were obtained. Is it possible to improve these estimates by changing the approximating subspace to another of the same dimension? And which estimates cannot be improved on the whole set of N-dimensional approximating subspaces?

Here, we are referring to the problem stated in Section 1.2 of finding the N-widths of functional classes M in a normed space X. The exact results from the previous chapters give upper bounds for the N-widths in the corresponding cases and now our attention will be concentrated on lower bounds for Kolmogorov N-widths. But a lower bound for the best approximations of the class M which is simultaneously valid for all N-dimensional approximating subspaces can only be obtained by using some very general and deep result. Borsuk's topological Theorem 2.5.1 stated in Section 2.5 turns out to be a suitable result in many cases.

In order to make the application of this theorem both possible and effective, one has to identify an (N + 1) parametric set MN+1 in M whose N-width is not less than the N-width of M. This depends on the metric of the space X and on the way of defining M. In the various cases the role of MN+1 may, for example, be played by a ball in subspaces of polynomials or splines, as some sets of perfect splines or their generalizations.