The Lyapunov stability theory for nonlinear time-varying dynamic system in Banach space is given in this paper. The Lyapunov stable theorem and the Barbashin-Krasovskii-LaSalle invariant set principle in classical theory are extended to infinite-dimensional Banach space. Under the assumptions of the existence of solution and the additive property of motions, sufficient and necessary conditions for uniform stability and uniform asymptotic stability are obtained, and the Lyapunov functions are explicitly constructed. This extension can be used as a criterion of stability for continuous and discontinuous systems.