Throughout my encounter with the mathematics of QM, I got used to the idea that observables are represented as self-adjoint operators in Hilbert spaces. Since self-adjoint operators have a real spectrum, this seems plausible. However, very often in quantum computation, I come across phrases like "doing a measurement in a given basis." Of course, every self-adjoint operator possesses a basis of eigenvectors (at least in the finite-dimensional setting). Bounded operators in general form a very rich and interesting mathematical structure (which is called a $C^*$-algebra and which is of its own independent interest): but if it is only the basis that matters, maybe one should give up working with operators and work with bases instead? However, if so, then I suspect that the particular ordering of the basis vectors does not matter, and the question arises:
What is the mathematical structure of the set of all bases in (finite-dimensional) Hilbert spaces?
If one insists on taking the ordering into account, then from every basis, one can form a unitary operator uniquely. But if we relax this requirement of keeping track of the ordering of the vectors, it is not clear to me what the right mathematical structure is when dealing with bases.
