Corollary

Suppose that the hilbertian space E is the direct sum of two orthogonal subspaces M and N. The canonical isomorphism g from $\Lambda(M) \otimes \Lambda(N)$ onto $\Lambda(E)$ (A, III, § 7, No. 7) extends in a unique way to a hilbertian space isomorphism from $\hat{\Lambda}(M) \hat{\otimes}_2 \hat{\Lambda}(N)$ onto $\hat{\Lambda}(E)$.

The proof is analogous to that of the corollary of prop. 4 (V, p. 31).

Let E and F be two hilbertian spaces and $u \in \mathscr{L}(E; F)$. We shall show, as in the case of symmetric powers $\hat{S}^n(E)$ (V, p. 32) that the linear mapping $\hat{\Lambda}^n(u)$ from $\hat{\Lambda}^n(E)$ into $\hat{\Lambda}^n(F)$ (A, III, § 7, No. 4) extends to a continuous linear mapping $\hat{\Lambda}^n(u)$ from $\hat{\Lambda}^n(E)$ into $\hat{\Lambda}^n(F)$. We have the relations

In general we do not have equality in formula (29) (TS, IV, § 6). Finally, we have an isomorphism $\psi_n = \psi_{n,E}$ from $\hat{\Lambda}^n(E)$ onto the subspace $\overline{\mathbf{A}}'_n(E)$ of $\hat{T}^n(E)$ defined by