Hilbert Space

"The literature on SVMs usually refers to the space H as a Hilbert space, so letís end this Section with a few notes on this point. You can think of a Hilbert space as a generalization of Euclidean space that behaves in a gentlemanly fashion. Specifically, it is any linear space, with an inner product defined, which is also complete with respect to the corresponding norm (that is, any Cauchy sequence of points converges to a point in the space). Some authors (e.g. (Kolmogorov, 1970)) also require that it be separable (that is, it must have a countable subset whose closure is the space itself), and some (e.g. Halmos, 1967) donít. Itís a generalization mainly because its inner product can be any inner product, not just the scalar (“dot”) product used here (and in Euclidean spaces in general). Itís interesting that the older mathematical literature (e.g. Kolmogorov, 1970) also required that Hilbert spaces be infinite dimensional, and that mathematicians are quite happy defining infinite dimensional Euclidean spaces. Research on Hilbert spaces centers on operators in those spaces, since the basic properties have long since been worked out. Since some people understandably blanch at the mention of Hilbert spaces, I decided to use the term Euclidean throughout this tutorial."
Burges (1998)

"DEFINITION 5. By a Hilbert space15 is meant a Euclidean space which is complete, separable and infinite-dimensional.

In other words, a Hilbert space is a set H of elements f, g,... of any kind such that

  1. H is a Euclidean space, i.e., a real linear space16 equipped with a scalar product;
  2. H is complete with respect to the metric ρ(f,g) = || f - g ||;
  3. H is separable, i.e., H contains a countable everywhere dense subset;
  4. H is infinite-dimensional, i.e., given any positive integer n, H contains n linearly independent elements.

Kolmogorov and Fomin, 1970