The problem of discrete-time and continuous-time adaptive stabilization under full-state feedback control is considered. In the discrete-time case the main result is based on a gain update law involving a step-size function. The formulation generalizes and unifies prior results based on quadratic and logarithmic Lyapunov functions. In the continuous-time case adaptive stabilization under full-state feedback using a normalized gradient algorithm is considered and Lyapunov stability is demonstrated.