  In a topological space, a family of continuous mappings is called universal
if its action, in at least one element of the space, is dense. If the mappings
are unitary or trace-preserving completely positive, the notion of universality
is closely related to the notion of controllability in either closed or open
quantum systems. Quantum controllability in infinite dimensions is discussed in
this setting and minimal generators are found for full control universal
families. Some of the requirements of the operators needed for control in
infinite dimensions follow from the properties of the infinite unitary group.
Hence, a brief discussed of this group and their appropriate mathematical
spaces is also included.
