Targeting at sparse learning, we construct Banach spaces of functions on an input space X with the following properties: (1) possesses an norm in the sense that is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on and are representable through a bilinear form with a kernel function; and (3) regularized learning schemes on satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.