Lemma

1— Let X and Y be two locally compact spaces, K (resp. L) a compact subset of X (resp. Y).

(i) The restriction to $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ of the canonical bijection

(S, R, §4, No. 14) is an isometry of the Banach space $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ onto the Banach space $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$.

(ii) The vector space $\mathscr{K}(X, K; \mathbf{C}) \otimes_{\mathbf{C}} \mathscr{K}(Y, L; \mathbf{C})$, identified canonically with a subspace of $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ (A, II, §7, No. 7, comments following the Cor. of Prop. 15), is dense in this Banach space.

It is immediate that the image under $\omega$ of $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ is contained in $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$. Conversely, if $u$ is a continuous mapping of $X$ into $\mathscr{K}(Y, L; \mathbf{C})$, with support contained in $K$, then the mapping $(x, y) \mapsto u(x)(y)$ of $X \times Y$ into $\mathbf{C}$ is continuous and has support contained in $K \times L$, therefore the restriction of $\omega$ to $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$ is a bijection of this space onto $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$; the fact that this restriction is a Banach space isometry follows from the relation

This proves (i); on the other hand the image under $\omega$, of

identified with a subspace of $\mathscr{K}(X \times Y, K \times L; \mathbf{C})$, is again the space $\mathscr{K}(X, K; \mathbf{C}) \otimes_{\mathbf{C}} \mathscr{K}(Y, L; \mathbf{C})$ but this time identified canonically with a space of mappings of $X$ into $\mathscr{K}(Y, L; \mathbf{C})$ (A, II, §7, No. 7, Cor. of Prop. 15); but this subspace of $\mathscr{K}(X, K; \mathscr{K}(Y, L; \mathbf{C}))$ is known to be dense in the latter space (§ 1, No. 2, Prop. 5), thus the conclusion of (ii) follows from the fact that the restriction of $\omega$ is a topological isomorphism.

Having proved the lemma, we now observe that every compact subset of $X \times Y$ is contained in a product $K \times L$, where $K$ (resp. $L$) is a compact subset of $X$ (resp. $Y$). It therefore follows from Lemma 1, (ii) that the subspace $\mathscr{K}(X; \mathbf{C}) \otimes_{\mathbf{C}} \mathscr{K}(Y; \mathbf{C})$ is dense in $\mathscr{K}(X \times Y; \mathbf{C})$; since the relation (1) may also be written as $\nu(g \otimes h) = \lambda(g)\mu(h)$ for $g \in \mathscr{K}(X; \mathbf{C})$, $h \in \mathscr{K}(Y; \mathbf{C})$, the uniqueness of $\nu$ follows at once. To prove the existence of $\nu$, we shall make use of the following lemma: