We now give the first application of the constructions of the previous chapter to prove a ‘prevalent’ version of a result first due to Mañé (1981). He showed that if X is a subset of a Banach space ℬ and dH(X − X) < k, then a residual subset of the space of projections onto any subspace of dimension at least k are injective on X.

We show here that in general no linear embedding into any ℝk is possible if we only assume that dH(X) is finite (Section 6.1). If we want an embedding theorem for such sets, we must fall back on Theorem 1.12 which guarantees the existence of generic embeddings of sets with finite covering dimension (we can apply this result since dim(X) ≤ dH(X) by Theorem 2.11).

While we prove in Theorem 6.2 the existence of a prevalent set of linear embeddings into ℝk when dH(X − X) < k, we will see that even with this assumption one cannot guarantee any particular degree of continuity for the inverse of the linear mapping that provides the embedding (Section 6.3).

In this chapter and those that follow, we will often wish to show that certain embedding results are sharp, in the sense that the information we obtain on the modulus of continuity for the inverse of the embedding map cannot be improved.