  We study condensation of trapped bosons in the limit when the number of
particles tends to infinity. For the noninteracting gas we prove that there is
no phase transition in any dimension, but in any dimension at any temperature
the system is 100% condensated into the one-particle ground state. In the case
of an interacting gas we show that for a family of suitably scaled pair
interactions, the Gross-Pitaevskii scaling included, a less-than-100%
condensation into a one-particle eigenstate, which may depend on the
interaction strength, persists at all temperatures.
