Continuum Quantum Systems as Limits of Discrete Quantum Systems, I: State Vectors

Dynamical systems on “continuum” Hilbert spaces may be realized as limits of dynamical systems on “discrete” (possibly finite-dimensional) Hilbert spaces. In this first of four papers on the topic, the “continuum” and “discrete” spaces are interfaced to one another algebraically, convergence of vectors is defined in such a way as to preserve inner products, and a necessary and sufficient coordinate-wise criterion for convergence is proved.