In Struwe's book, the section on Ekeland's variational principal, i.e, Thm. 5.1 has that $(M, d)$ is a metric space in its hypothesis. In Thm. 5.5, he uses Thm. 5.1 with \begin{equation} M\equiv u_0+H^{1, 1}_0(\Omega;\mathbb{R}^N) \end{equation}and \begin{equation} d(w, v)\equiv \|Dw-Dv\|_{L^1(\Omega)}. \end{equation} Here $u_0$ is any element in $H^{1, s}(\Omega;\mathbb{R}^N)$ and $s\in (1, \infty)$.
If $u_0$ does not have zero trace, then why is $u_0+H_0^{1, 1}(\Omega;\mathbb{R}^N)$ a metric space with respect to the metric $d$? Doesn't $0$ have to belong to $M$ for $M$ to be a metric space?