We give examples on the use of the Stone-Weierstrass theorem in inverse problems. The theorem allows us to generalize and simplify earlier results. We also show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds, and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. The Stone-Weierstrass theorem makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.