If $\mathcal{T}_h$ is a regular and quasi-uniform triangulation of $\Omega$, and $V_h$ is the $H^1$-conforming linear finite element space. Moreover, let $P_h$ be the $L^2$ projection to $V_h\subset H^1(\Omega)$, i.e., \begin{equation} (P_hu,v_h)=(u,v_h),\quad\quad \forall v_h\,\in V_h\quad\quad (1). \end{equation} How to show the stability of the following result \begin{equation} \|P_hu\|_{H^1(\Omega)}\leq C\|u\|_{H^1(\Omega)}\quad\quad (2). \end{equation}
By choosing $v_h=P_hu$, we have \begin{equation} \|P_hu\|_{L^2(\Omega)}^2=(u,p_hu)\leq \|u\|_{L^2(\Omega)}\|P_hu\|_{L^2(\Omega)}, \end{equation} which shows that \begin{equation} \|P_hu\|_{L^2(\Omega)}\leq \|u\|_{L^2(\Omega)}. \end{equation} But, how to show the result $(2)$? Could anyone give some suggestions? Thank you!