This paper proposes an iterative learning control (ILC) scheme for a class of redundant robot arms to acquire the desired control input signals that produce an endpoint trajectory specified in task space. The learning update law of control input signals is constructed only in task space by modifying the previous control input through adding linearly an endpoint velocity trajectory error. Although the dimension of the task space is strictly less than the DOF (degrees-of-freedom) of the robot arm, the proposed method need neither consider any inverse kinematics problem nor introduce any cost function to be optimized and to determine the inverse kinematics (or dynamics) uniquely. Convergence of trajectory trackings to the specified one is shown by numerical simulations in both cases 1) free-endpoint motion and 2) constraint-endpoint motion with specified contact force. A theoretical proof of convergences in the case of free-endpoint motion is given on the basis of an approximated dynamics linearized around a desired solution in joint state space.