Pour-El & Richards  discussed an ad hoc computability structure in an effectively separable Hilbert space taking as an effective generating set a slightly modified one from the original orthonormal basis. We show that an application of the Poincaré-Wigner orthogonalizing procedure to Pour-El & Richards' modified system gives an orthonormal effective generating set which yields a third computability structure.
"On an ad hoc computability structure in a Hilbert space." Proc. Japan Acad. Ser. A Math. Sci. 79 (3) 65 - 70, March 2003. https://doi.org/10.3792/pjaa.79.65