Example 3

Tag 07XJ, permanent. Cite it as [Bourbaki, Tag 07XJ].

Topological Vector Spaces, chapter V, § 1. Prehilbertian spaces and hilbertian spaces

\* Let $n \geqslant 1$ be an integer and let U be an open set in $\mathbf{R}^n$ . Let $\mu$ be the measure on U induced by the Lebesgue measure on $\mathbf{R}^n$ , and put $\mathscr{H}^0 = L^2(U, \mu)$ . Let $\mathscr{H}^1$ denote the space of all functions $f \in \mathscr{H}^0$ with the following property; for $1 \leqslant i \leqslant n$ , there exists a function $g_i \in \mathscr{H}^\circ$ such that

\int_U g_i(x)\ h(x)\ d\mu(x) = - \int_U f(x)\ D_i h(x)\ d\mu(x)

for every function $h$ of class $C^1$ with compact support in U. The function $g_i$ is defined uniquely up to equivalence with respect to $\mu$ , and is denoted by $D_i f$ or $\partial f / \partial x_i$ (ith partial derivative). By induction on the integer $s \geqslant 1$ , we define $\mathscr{H}^s$ as the set of all functions $f \in \mathscr{H}^1$ such that $D_i f \in \mathscr{H}^{s-1}$ for $1 \leqslant i \leqslant n$ . We define a scalar product on $\mathscr{H}^s$ by the formula

\langle f|g \rangle = \sum_{k=0}^s \sum_{1 \leqslant i_1 \leqslant \ldots \leqslant i_k \leqslant n} \int \overline{D_{i_1} \ldots D_{i_k} f} \cdot D_{i_1} \ldots D_{i_k} g\ d\mu .

Then $\mathscr{H}^s$ is a complex hilbertian space, called Sobolev space of index s.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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