Proposition 6

Let E be a hilbertian space. We have

for $x \in E$ and $u \in \hat{\Lambda}(E)$.

It is clearly enough to consider the case $\| x \| = 1$.

Let F be the hilbertian subspace of E consisting of all vectors orthogonal to x. Since E is the hilbertian sum of F and the line K.x, it follows from the corollary of V, p. 34 that the mapping $(v, w) \mapsto v + x \wedge w$ is a hilbertian space isomorphism from $\hat{\Lambda}(F) \oplus \hat{\Lambda}(F)$ onto $\hat{\Lambda}(E)$. If $u = v + x \wedge w$ with $v, w$ in $\hat{\Lambda}(F)$, we have $x \wedge u = x \wedge v$, hence $\| x \wedge u \| = \| v \| \leq (\| v \|^2 + \| w \|^2)^{1/2} = \| u \|$.