  We first briefly review some aspects of the techniques of dealing with
ultraviolet divergences in Feynman amplitudes in an Euclidian $D$-dimensional
space-time. Next we consider compactification of a $d$-dimensional ($d\leq D$)
subspace. This includes effects of temperature and of compactification of $d-1$
spatial coordinates. Then we show how dimensional renormalization can be
implemented for a field theory defined on this Euclidian space-time with a
compactified subspace.
