We investigate the procedures of discretization of the integrable nonlinear Schrodinger dynamical system, well known as the Ablowitz–Ladik equation, the corresponding symplectic structures, and the finite-dimensional invariant reductions. We develop an efficient scheme of invariant reduction of the corresponding infinite system of ordinary differential equations to an equivalent finite system of ordinary differential equations with respect to the evolution parameter. We construct a finite set of recurrence algebraic regular relations that allows one to generate solutions of the discrete nonlinear Schrodinger dynamical system and discuss the related functional spaces of solutions. Finally, we analyze the Fourier-transform approach to the study of the set of solutions of the discrete nonlinear Schrodinger dynamical system and its functional-analytic aspects.