  We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$,
for which the standard functional $$ ||f||_{p,s}=(\int_0^\infty
(t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s}
  $$ is only a quasi-norm. We find the optimal constant in the triangle
inequality for this quasi-norm, which leads us to consider the following
decomposition norm: $$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\},
$$ where the infimum is taken over all finite representations $f=\sum_{k}f_k. $
We also prove that the decomposition norm and the dual norm $$ ||f||_{p,s}'=
\sup\left\{\int_R fg d\mu: ||g||_{p',s'}=1\right\} $$ agree for all values
$p,s>1$.
