Example

Let U be an open set of $\mathbf{R}^n$ endowed with the Lebesgue measure $\mu$. Let Δ be the scalar differential operator

on U. Let us denote by $u$ the partial operator on the real Hilbert space $L^2_{\mathbf{R}}(U, \mu)$ with domain $\mathscr{D}_{\mathbf{R}}(U) ($IV, p. 201, remark) defined by $\varphi \mapsto -\sum^n_{i=1}\partial_i^2\varphi$. One has $u_{(\mathbf{C})}= \Delta_-$, which is a closable partial operator (prop. 13 of IV, p. 242) and symmetric (since Δ is formally symmetric), hence $u$ is closable and symmetric (proposition 2 of IV, p. 228 and lemma 3 of IV, p. 239). By the above corollary, there therefore exists a self-adjoint extension of $u_{(\mathbf{C})}$. It is a Laplacian on U (IV, p. 243).