Remark

The proposition means that the adjoint of a partial operator $u$ such that $D_-\subset u\subset D_+$ can always be calculated in the sense of distributions: the elements $f$ of the domain of $u^*$ are elements of $L^2(U)$ such that the distribution $^tD(f)$ belongs to $L^2$(U), and one has $u^*(f) =^tD(f)$.

One says that D is formally symmetric if $^tD = D$ as a scalar differential operator. If this is the case, the partial operator $D_-$ is symmetric.

Consider the particular case of the scalar differential operator of order 2 defined by

A Laplacian on U is any partial operator $u$, self-adjoint on $L^2$(U), such that $\Delta_-\subset u($cf. VAR, R2, 14.4.3, p. 83). We shall see later (IV, p. 261, example) that there always exists at least one Laplacian on $L^2(U)$; there may exist more than one (exercise 17 of IV, p. 358).