  In this paper we characterize sparse solutions for variational problems of
the form $\min_{u\in X} \phi(u) + F(\mathcal{A} u)$, where $X$ is a locally
convex space, $\mathcal{A}$ is a linear continuous operator that maps into a
finite dimensional Hilbert space and $\phi$ is a seminorm. More precisely, we
prove that there exists a minimizer that is `sparse' in the sense that it is
represented as a linear combination of the extremal points of the unit ball
associated with the regularizer $\phi$ (possibly translated by an element in
the null space of $\phi$). We apply this result to relevant regularizers such
as the total variation seminorm and the Radon norm of a scalar linear
differential operator. In the first example, we provide a theoretical
justification of the so-called staircase effect and in the second one, we
recover the result in [31] under weaker hypotheses.
