Postscript: Infinite Versus Finite Dimensions

Abstract

We have chosen to finish this book by indicating many of the ways in which finite dimensionality has played a critical role in the previous chapters. While our list is far from complete it should help illuminate the places in which care is appropriate when “generalizing”. Many of our main results (on subgradients, variational principles, open mappings, Fenchel duality, metric regularity) immediately generalize to at least reflexive Banach spaces. When they do not, it is principally because the compactness properties and support properties of convex sets have become significantly more subtle. There are also significantly many properties that characterize Hilbert space. The most striking is perhaps the deep result that a Banach space X is (isomorphic to) Hilbert space if and only if every closed vector subspace is complemented in X. Especially with respect to best approximation properties, it is Hilbert space that best captures the properties of Euclidean space.