The paper deals with the weakly-singular linear Volterra integral equation of the second kind
The unique solution of the equation belongs to which is proved to be a reproducing kernel space with simple reproducing kernel. The expression of the reproducing kernel is given. The reproducing kernel method of a linear operator equation , which requests the image space of the operator is and the operator is bounded, is improved. Namely, the request for the image space is weakened to be and the boundedness of the operator is also not required. The authors give the exact solution of the equation, denoted by a series in the reproducing kernel space After truncating the series, the approximate solution (which converges uniformly to the exact solution) is obtained. The effectiveness of the method is shown by the numerical experiments.