We consider the problem of recovering conditional independence relationships between p jointly distributed Hilbertian random elements given n realizations thereof. We operate in the sparse high-dimensional regime, where n ≪ p and no element is related to more than d ≪ p other elements. In this context, we propose an infinite-dimensional generalization of the graphical lasso. We prove model selection consistency under natural assumptions and extend many classical results to infinite dimensions. In particular, we do not make additional structural assumptions. The plug-in nature of our method makes it applicable to heterogeneous data measured under any observational regime, whether sparse or dense, and indifferent to serial dependence between samples. In addition, it does not require dimensionality reduction by truncation. Importantly, our method can be understood as naturally arising from a coherent maximum likelihood philosophy.