Is the cardinality of the dimension of an infinite-dimensional vector space well-defined?

Question

I thought of this question because of something in physics. In quantum mechanics, the state of a system is associated with a unit vector in a Hilbert space, which in many cases can be thought of as $L^2$. Physicists often project this vector to different bases for convenience. However, sometimes the basis is countable and other times it is not, which looks very suspicious. Usually, when this happens, the uncountable basis is the Dirac delta functions (and maybe we could think of the countable basis as the Hermite polynomials or something). The Dirac delta functions are not functions at all so this is nowhere rigorous in the first place, but, could this phenomenon actually happen in math?

Answer 1

1. Yes, the dimension of an infinite-dimensional vector space is well-defined, in the sense that any two bases have the same cardinality.

2. For Hilbert spaces, however, the dimension is not the correct invariant; instead you want the cardinality of an orthonormal basis, which is not a basis in the algebraic sense (a Hamel basis) because one uses infinite sums and not just finite sums. Hilbert spaces have a "Hilbert space dimension" which is also well-defined, namely the cardinality of an orthonormal basis. Up to isomorphism there is a unique separable Hilbert space, which is the unique Hilbert space with a countable orthonormal basis; e.g. $\ell^2(\mathbb{Z}), L^2(S^1), L^2(\mathbb{R}^n)$ are all separable and so all isomorphic as Hilbert spaces and all have the same Hilbert space dimension, which is countable. Their algebraic dimension is uncountable but this is irrelevant in practice.

3. The use of delta functions to represent elements of, say, $L^2(\mathbb{R})$, of which there are uncountably many, does not contradict the fact that $L^2(\mathbb{R})$ has countable Hilbert space dimension, because the delta functions 1) are not even in $L^2(\mathbb{R})$, and 2) the way they are used to build up functions is through taking integrals, not infinite sums and not finite sums. The way to understand this phenomenon formally in mathematical terms is using what is called a direct integral.