Hilbert Spaces with Generic Predicates

We study the model theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic expansions of Hilbert spaces in the form first of a distance function to a random substructure, then a distance to a random subset. The theory obtained with the random substructure is {\omega}-stable, while the one obtained with the distance to a random subset is $TP_2$ and $NSOP_1$. That example is the first continuous structure in that class.