The linear isometry group of the Gurarij space is universal

We give a construction of the Gurarij space, analogous to Katetov's construction of the Urysohn space. The adaptation of Katetov's technique uses a generalisation of the Arens-Eells enveloping space to metric space with a distinguished normed subspace. This allows us to give a positive answer to a question of Uspenskij, whether the linear isometry group of the Gurarij space is a universal Polish group.