  Let $\M$ be a semi-finite factor and let $\J(\M)$ be the set of operators $T$
in $\M$ such that $T=ETE$ for some finite projection $E$. In this paper we
obtain a representation theorem for unitarily invariant norms on $\J(\M)$ in
terms of Ky Fan norms. As an application, we prove that the class of unitarily
invariant norms on $\J(\M)$ coincides with the class of symmetric gauge norms
on a classical abelian algebra, which generalizes von Neumann's classical
result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As another
application, Ky Fan's dominance theorem \cite{Fan} is obtained for semi-finite
factors. Some classical results in non-commutative $L^p$-theory (e.g.,
non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of
non-commutative $L^p$-spaces) are extended to general unitarily invariant norms
related to semi-finite factors. We also prove that up to a scale the operator
norm is the unique unitarily invariant norm associated to a type ${\rm III}$
factor.
