  We give examples on the use of the Stone-Weierstrass theorem in inverse
problems. We show uniqueness in the linearized Calder\'on problem on
holomorphically separable K\"ahler manifolds, and in the Calder\'on problem for
nonlinear equations on conformally transversally anisotropic manifolds. We also
study the holomorphic separability condition in terms of plurisubharmonic
functions. The Stone-Weierstrass theorem allows us to generalize and simplify
earlier results. It also makes it possible to circumvent the use of complex
geometrical optics solutions and inversion of explicit transforms in certain
cases.
