  We consider the problem of constructing a linear map from a Hilbert space
$\mathcal{H}$ (possibly infinite dimensional) to $\mathbb{R}^m$ that satisfies
a restricted isometry property (RIP) on an arbitrary signal model $\mathcal{S}
\subset \mathcal{H}$. We present a generic framework that handles a large class
of low-dimensional subsets but also unstructured and structured linear maps. We
provide a simple recipe to prove that a random linear map satisfies a general
RIP on $\mathcal{S}$ with high probability. We also describe a generic
technique to construct linear maps that satisfy the RIP. Finally, we detail how
to use our results in several examples, which allow us to recover and extend
many known compressive sampling results.
