Let (S, ∥·∥) be a Banach space (in general nonseparable). A subset of the unit ball {f ∈ S′ : ∥f∥′ ≤ 1} is called a norming subset if and only if for all s ∈ S. The whole unit ball in S is always a norming subset by the Hahn-Banach theorem (RAP, Corollary 6.1.5).

Conversely, given any set, let be the set of all bounded real functions on, with the supremum norm

Then the natural map f ↦ (s ↦ s(f)) takes one-to-one onto a norming subset of S′.

So, limit theorems for empirical measures, uniformly over a class of functions, can be viewed as limit theorems in a Banach space S with norm Conversely, limit theorems in a general Banach space S with norm ∥ · ∥ can be viewed as limit theorems for empirical measures on S, uniformly over a class of functions, such as the unit ball of S′, since for f ∈ S′ and x1, …, xn ∈ S,

Suppose that Xj are i.i.d. real random variables with mean 0 and variance 1. Let One form of “invariance principle” will say that on some probability space, there exist such Xj and also i.i.d. N(0, 1) variables Y1, Y2, …, with such that as n → ∞ in probability. Since Tn/n½ also has a N(0, 1) distribution for each n, the invariance principle implies that Sn/n½ is close to Tn/n½, which implies the central limit theorem. Although it is not as obvious, central limit theorems generally imply invariance principles.