Proposition 11

Tag 02R8, permanent. Cite it as [Bourbaki, Tag 02R8].

Théories spectrales, chapter III, § 2. Exemples d’applications linéaires compactes

Suppose that the vector bundle E is locally of finite rank. Let Y be an open relatively compact subset of X. The restriction mapping $f\mapsto f|Y$ of $\mathscr{S}^{\omega}(X; E)$ into $\mathscr{S}^{\omega}(Y; E)$ is compact.

With the notation of lemma 5, one has a commutative diagram

\mathscr{S}^{\omega}(X; E)\leftarrow_i\rightarrow \mathscr{S}^1(X_0; E_0)

\rightarrow \leftarrow_u\rightarrow \leftarrow_v \tag{3}

\mathscr{S}^{\omega}(Y; E)\leftarrow_j\rightarrow \mathscr{S}^0(Y_0; E_0)

where $i$ and $j$ are the canonical injections and $u,v$ the restriction mappings. The mapping $i$ is continuous (Lemma 5, b)), and the mapping $u$ is compact (Proposition 10). Since the canonical injection $j$ is continuous, strict, and has closed image (Lemma 5, a)), the mapping $u$ is compact (Remark 5 of III, p. 3).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.