In What is the Computational Basis? gIS states:
One also often speaks of "computational basis" for higher-dimensional states (qudits), in which case the same applies: a basis is called "computational" when it's the most "natural" in a given context.
The Wikipedia page for Hilbert space includes this snippet:
When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.
If the Hilbert space has countably infinite axes, is the computational basis for it transfinite?
Additionally, would it be accurate to state that computations with finite bases correlate to classical computations, while computations with transfinite bases correlate quantum computations?