  We provide explicit formulas for the norm of bounded linear functionals on
Orlicz-Lorentz function spaces $\Lambda_{\varphi,w}$ equipped with two standard
Luxemburg and Orlicz norms. Any bounded linear functional is a sum of regular
and singular functionals, and we show that the norm of a singular functional is
the same regardless of the norm in the space, while the formulas of the norm of
general functionals are different for the Luxemburg and Orlicz norm. The
relationship between equivalent definitions of the modular $P_{\varphi,w}$
generating the dual space to Orlicz-Lorentz space is discussed in order to
compute the norm of a bounded linear functional on $\Lambda_{\varphi,w}$
equipped with Orlicz norm. As a consequence, we show that the order-continuous
subspace of Orlicz-Lorentz space equipped with the Luxemburg norm is an
$M$-ideal in $\Lambda_{\varphi,w}$, while this is not true for the space with
the Orlicz norm when $\varphi$ is an Orlicz $N$-function not satisfying the
appropriate $\Delta_2$ condition. The analogous results on Orlicz-Lorentz
sequence spaces are given.
