I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable. But in QM it's countable... because everything comes in multiples of Planck's constant, perhaps? Though I haven't seen the actual reason stated, and perhaps it's something beyond my current grasp.

Ahh... here's something I can help with. To see why Hilbert space has a countable basis, let's first define Hilbert space. So let

= the set of all functions such that the integral of is finite, and let

= the set of all functions such that the integral of is zero. This includes for example the Dirichlet function which is one on rational numbers but zero on irrational numbers. So it's actually a pretty big space.

Hilbert space is defined to be the quotient space . To see that it has a countable basis, it suffices to show that it contains a countable dense set. Then the Gram-Schmidt orthogonalization process can turn that set into a basis. What does it mean to say that a set is dense? Well, the metric on Hilbert space is given by the formula

,

so a sequence is dense if for every element of Hilbert space, you can find a sequence such that . Now we can see why we needed to mod out by -- any two points of are considered to have distance zero from each other!

So what's a countable dense sequence? One sequence that works is the sequence of all piecewise-linear continuous functions with finitely many pieces whose vertices are rational numbers. This class includes for example the function defined by the following equations:

for all

for all

for all

for all

Note that I don't need to specify what does if I plug in a number in the finite set , since any function which is zero outside of that set is an element of , so would represent the same element of Hilbert space as .

So to summarize:

1. The uncountable set that you would intuitively think is a basis for Hilbert space, namely the set of functions which are zero except at a single value where they are one, is in fact not even a sequence of distinct elements of Hilbert space, since all these functions are elements of , and are therefore considered to be equivalent to the zero function.

2. The actual countable basis for Hilbert space will look much different, and the Gram-Schmidt process I alluded to above doesn't really let you say exactly what the basis looks like. For Hilbert space over the unit interval, there is a convenient way to get around this, namely Parseval's theorem, which states that the sequences and form a basis for Hilbert space. For Hilbert space over the entire real line, there are some known bases but they aren't as elegant, and in practice we rarely need an explicit countable basis.

3. Finally, the philosophical aspect: Having a countable basis means that elements of Hilbert space can be approximated arbitrarily well by elements which take only a finite amount of information to describe*, much like real numbers can be approximated by rational numbers. This means that an infinite set atheist should be much more comfortable with countable-basis Hilbert space than with uncountable-basis Hilbert space, where such approximation is impossible.

* The general rule is:

Elements of a finite set require a finite and bounded amount of information to describe.

Elements of a countable set require a finite but unbounded amount of information to describe.

Elements of an uncountable set (of the cardinality of the continuum) require a countable amount of information to describe.