A computational method is proposed in this paper for a minimax problem in the time domain. A minimax problem for a linear system is formulated and analyzed by vector space methods. The uniqueness of the optimal solution is analyzed readily in the present approach, and its geometric interpretation is presented. The further analysis based on duality shows that the infimum or a lower bound of the value of the performance index is calculated through supplementary optimization problems, and the minimax problem is solved as an optimization problem with an inequality constraint. The proposed approach provides a simple computational method for the minimax problem. Two numerical examples demonstrate simple implementation of the proposed method.