The term ‘prevalence’ was coined by Hunt et al. (1992), for a generalisation of the notion of ‘almost every’ that is appropriate for infinite-dimensional spaces. Essentially the same definition was used earlier by Christensen (1973), although for him a set was prevalent if its complement was a Haar null set; we adopt here the more recent and more descriptive terminology. A nice review of the theory of prevalence is given by Ott & Yorke (2005). We only develop the theory here as far as we will need it in what follows; more details can be found in the above papers and in Benyamini & Lindenstrauss (2000, Chapter 6).

Once we have introduced prevalence, we show how the idea can be adapted to treat certain classes of linear maps from infinite-dimensional spaces into finite-dimensional Euclidean spaces (Section 5.2), and then prove a generalisation of the inequality (4.1) that is a key element of the subsequent embedding proofs.

Prevalence

Let V be a normed linear space. First we define what it means for a subset of V to be ‘shy’, the equivalent in this setting of ‘having measure zero’; the complement of a shy set is said to be ‘prevalent’.