Lemma 2

Let $n\in \mathbf{N}$. The mapping $\widehat{\wedge}^ni_H$ is an isometric linear mapping of $\widehat{\wedge}^nH$ into $\widehat{\wedge}^nE$.

Let $(e_j)_{j\in J}$ be an orthonormal basis of H and $(e_j)_{j\in J'}$ an orthonormal basis of E, where $J\subset J'$. Let us endow $J'$ with a total ordering. The elements $e_{j_1}\wedge \cdots \wedge e_{j_n}$ for $j_1<\cdots < j_n$ in $J'$ (resp. in J) form an orthonormal basis of $\widehat{\wedge}^nE$ (resp. of $\widehat{\wedge}^nH$) by Prop. 5 of EVT, V, p. 34, prop. 5. The lemma follows.

In what follows, we shall identify $\widehat{\wedge}^nH$ with a closed subspace of $\widehat{\wedge}^nE$ by means of the mapping $\widehat{\wedge}^ni_H$.