Let

- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be open
- $\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$ and $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\|_{L^2(\Lambda,\:\mathbb R^d)}}$$
- $\operatorname P_H$ denote the orthogonal projection from $L^2(\Lambda,\mathbb R^d)$ onto $H$

In Remark 1.6 of *Navier-Stokes Equations: Theory and Numerical Analysis* by *Roger Temam*, the author is stating that $\text P_HH_0^1(\Lambda,\mathbb R^d)\subseteq H_0^1(\Lambda,\mathbb R^d)$. I don't think that this is trivial. How can we prove it?

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