Summary: Targeting at sparse learning, we construct Banach spaces \(\mathcal B\) of functions on an input space \(X\) with the following properties: (1) \(\mathcal B\) possesses an \(\ell ^{1}\) norm in the sense that \(\mathcal B\) 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 \(\mathcal B\) and are representable through a bilinear form with a kernel function; and (3) regularized learning schemes on \(\mathcal B\) satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.