Complete set of commuting observables

Remarks

By clause 1, the $A_i$ are commuting self-adjoint operators, so they have a common orthonormal basis of eigenvectors. Clause 2 says that this basis is unique up to order and phases, and that each basis vector is labelled by its tuple of eigenvalues. Equivalently, every operator that commutes with all $A_i$ is a polynomial in $A_1, \dots, A_k$. In infinite dimension, one asks that the spectral projections of the $A_i$ commute and that the von Neumann algebra they generate be maximal abelian.

Example: let $e_1, e_2$ be the standard basis of $\C^2$. On $\C^2 \otimes \C^2 \cong \C^4$, let $A_1 = \sigma_z \otimes I$ and $A_2 = I \otimes \sigma_z$ with $\sigma_z = \mathrm{diag}(1, -1)$. They commute, and the four joint eigenspaces for $(\pm 1, \pm 1)$ are spanned by the four vectors $e_a \otimes e_b$, each of dimension $1$.

Non-example: $\set{A_1}$ alone commutes with itself, but its eigenspace for the eigenvalue $1$ is spanned by $e_1 \otimes e_1$ and $e_1 \otimes e_2$, of dimension $2$, so clause 2 fails.