10 - Further remarks and notes

Summary

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

(Carl Friedrich Gauss)

Back to the finite

We finish this book by reprising some 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 results have effectively the same proof in Banach spaces as they do in Euclidean spaces.

The general picture

For example, the equivalence of local boundedness and local Lipschitz properties for convex functions is a purely geometric argument that does not depend on the dimension of the space (compare Theorem 2.1.10 and Proposition 4.1.4). Consequently, in finite dimensions a real-valued convex functions is automatically continuous essentially because a simplex has nonempty interior (see Theorem 2.1.12). This argument clearly fails in infinite-dimensional spaces, and the result fails badly as there are discontinuous linear functionals on every infinite-dimensional Banach space (Exercise 4.1.22).