When there is no end to the process of selecting a basis for a vector space we have an infinite dimensional space.
Much of the application of vector spaces involves linear combinations of basis vectors.
The definition of a basis in an n dimensional space speaks of the sum of n vectors.
For an infinite dimensional space we need to define the sum of a denumerable set of vectors and this requires some concept of continuity, convergence or other topological construct.
Some sums will not converge and a good deal of unruly hair enters the scene.
David Hilbert arranged that hair beautifully.
For instance the vector space of functions defined on the real line would seem to be of dimension c since there are that many possible arguments to such a function.
Summing over a denumerable set is hairy—over a non-denumerable set much worse.
The idea of quadratic form also seems out of reach at least in its concrete description.
Functions from a finite domain to the reals or complex numbers, form a natural vector space; each element x of the domain corresponds to the function `f(z) = 1 if z=x else 0` and these functions compose a basis for the vector space.
When the domain is the reals this leads to a c dimensional vector space which seems to lack any sort of quadratic form; you can’t sum c reals, even with a limiting notion.

If we want a vector space that includes the functions that we find useful in physics, for example, we would want to include g = λx.e^{−x2} and assign a magnitude to it just as quadratic forms assign magnitudes to all vectors.
As for functions like [f(x) = 1 if x=0 else 0] we must either exclude it from the vector space or put it in an equivalence class with [f(x) = 0].

With some artful triage David Hilbert sorted these things out and selected a class of useful and civilized functions that could form a powerful vector space later called “Hilbert space”.
More specifically he selected a set of complex valued functions defined over an arbitrary measure space and defined equivalence classes of such functions.
If you don’t know “measure space” think real line and you won’t miss much here.

Hilbert space has c points.

### Detritus

#### Axioms for Hilbert Space

There are simple axioms for Hilbert space.
You start with the axioms for an ordinary vector space with a positive definite quadratic form which provides a metric, and further postulate that the process of finding a basis does not end—that any basis is thus infinite.
Another axiom is that the space is **complete** which means that any Cauchy sequence of points converges to a point in the space.
Then you add an axiom that says that the space is **separable** which means that there is a denumerable subset of the space that is dense in the space.
A subset D is dense in X when for any point x in X and any positive real z, there is a point in D within distance z of x.
The infinite basis makes the space big enough and it turns out that separability keeps it from being too big.
If you fix the field of the vector space (presumably either reals or complex) then any two Hilbert spaces are isomorphic to each other.
Here are two concrete constructions of Hilbert space:

- Each vector in the space is an infinite sequence of complex numbers with the requirement that the sum of the squared magnitudes converge.
This sum is the quadratic form.
The dot product of two such vectors converges.
- Each vector is an equivalence class of complex functions on the real interval [0, 1].
The integral of the square of their magnitude must converge.
Two functions are equivalent if where they differ is a set of measure zero.

These two spaces are isomorphic which is the precise form of what Fourier discovered.
I think Hilbert dealt only with separable spaces but inseparable spaces are now included which are much bigger and require non constructive proofs.

I think this is a good intro to Hilbert spaces.