In 1953 Yu. Prohorov published a paper on weak convergence of
probability measures on metric spaces, bringing a new, extended context to the Invariance Principle proved by Donsker two years earlier.
Prohorov’s formalism, publicised in books by K.R. Parthasarathy and
P. Billingsley, established the equivalence of the notion of convergence
in law of stochastic processes and the weak convergence of their distributions. This point of view is completely justified in metric spaces,
especially in Polish spaces.

It is, however, much less satisfactory in non-metric spaces, as was
shown by examples due to X. Fernique, given long time ago.
We show that in a large class of submetric spaces there exists a
stronger mode of convergence, coinciding with the weak convergence
on metric spaces, and much more suitable for needs of contemporary
theory of stochastic partial differential equations.

A submetric space is a topological space (X,tau) admitting a continuous metrics d that in turn determines a metric topology $\tau_d \subset \tau$ (where this inclusion is in general strict). As a standard (and the simplest)
example may serve a separable Hilbert space equipped with the weak
topology.