How to show the stability of $L^2$ projection?

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., $$$$(P_hu,v_h)=(u,v_h),\quad\quad \forall v_h\,\in V_h\quad\quad (1).$$$$ How to show the stability of the following result $$$$\|P_hu\|_{H^1(\Omega)}\leq C\|u\|_{H^1(\Omega)}\quad\quad (2).$$$$

By choosing $$v_h=P_hu$$, we have $$$$\|P_hu\|_{L^2(\Omega)}^2=(u,p_hu)\leq \|u\|_{L^2(\Omega)}\|P_hu\|_{L^2(\Omega)},$$$$ which shows that $$$$\|P_hu\|_{L^2(\Omega)}\leq \|u\|_{L^2(\Omega)}.$$$$ But, how to show the result $$(2)$$? Could anyone give some suggestions? Thank you!

• Can you clarify what you mean by "how to show result (1)?" This is the definition of the projection. Commented Jun 17, 2019 at 15:17

I'd assume you want to prove $$(2)$$, and $$u\in V\subset H^1(\Omega)$$.
You already proved that $$\|P_hu\|_{L^2(\Omega)}\leq \|u\|_{L^2(\Omega)}$$, which is just one more step to arrive $$\|P_hu\|_{H^1(\Omega)}\leq C_0\left(\|u\|_{H^1(\Omega)}+\|u\|_{L^2(\Omega)}\right).$$ Then you prove $$\|u\|_{L^2(\Omega)} \leq C_1\|u\|_{H^1(\Omega)}.$$
Combine the above two inequalities, you get $$\|P_hu\|_{H^1(\Omega)}\leq C\|u\|_{H^1(\Omega)}.$$