In order to apply Propositions 13.1, 13.2, and 13.3, we need a useful criterion that tells us when a series of functions converges uniformly over a set S. Weierstrass gave such a criterion for …

Intuitively, gn(x) → g(x) uniformly if it is possible to draw an -band around the graph of g(x) that contains all of the graphs of gn(x) for large enough n.

This example shows that a function can be uniformly contin-uous on a set even though it does not satisfy a Lipschitz inequality on that set, i.e. the method of Theorem 8 is not the only method …

The functions fi are all continuous functions on a compact set (hence uniformly continuous by Theorem 1.2). If fi ! f uniformly, then f is continuous, however we know that the limit f is not …

To illustrate the use of this new characterization of uniform con-tinuity, we reprove one of the standard theorems (see e.g. [Roy88]) about uniformly continuous functions:

Theorem. The series P1 n=1 un(z) is uniformly convergent on a compact subset only if the following holds. For each > 0 there is a value n0 depending on R such that the p rtial sums Sm z)

Theorem 8 (Milman–Pettis) If X is uniformly convex, then X is reflexive, i.e. X∗∗ = X.

Every function that’s uniformly continuous on a dense subset has a continuous extension to the whole set. To make this statement precise, let’s recall that, for a set A of real numbers, a …

vergence of Functions Section 24 Let (fn) be a sequence of real valu. d functions all with domain D R. Let f be a real . alued function with do-main D R. We want to de ne what it . eans for (fn) …

Since P1 n=1Kn is a geometric series with 0 < 1, it converges which implies P1 n=1 5n2ntn 2 converges uniformly on [ 2+r;2 r] to a function U. Since limits are unique, we have S = U on [ …