Exercise 2, § 3

Let E be an infinite dimensional hilbertian space, and let $p, q$ be two integers $\geq 1$; let $p' = \left[ \frac{p}{2} \right], q' = \left[ \frac{q}{2} \right]$ (integral parts). Show that the norm of the bilinear mapping $(u, v) \mapsto u \wedge v$ from $\hat{\Lambda}^p(E) \times \hat{\Lambda}^q(E)$ into $\hat{\Lambda}^{p+q}(E)$ is at least equal to $\left( \frac{(p'+q')!}{p'! \ q'!} \right)^{1/2}$. (When $p = 2p'$ and $q = 2q'$ are even, consider a 2n-dimensional subspace $E_n$ in E, with an orthonormal basis $(e_j)_{1 \leq j \leq 2n}$; let $e'_j = e_{2j-1} \wedge e_{2j}$ for $1 \leq j \leq n$; consider the product $u \wedge v$, where $u = \sum_H e'_H$, $v = \sum_{\mathbf{K}} e'_\mathbf{K}$ where $\mathbf{H}$ (resp. $\mathbf{K}$) range over the set of all subsets of $p'$ (resp. $q'$) elements of $\{1, 2, ..., n\}$ and $e'_\mathbf{H} = e'_{i_1} \wedge ... \wedge e'_{i_p}$ (resp. $e'_\mathbf{K} = e'_{j_1} \wedge ... \wedge e'_{j_q}$) if $i_1 < ... < i_p$ (resp. $j_1 < ... < j_q$) is the increasing sequence of elements of $\mathbf{H}$ (resp. $\mathbf{K}$). Deduce that the multiplication in $\Lambda(\mathbf{E})$ cannot be extended to $\hat{\Lambda}(\mathbf{E})$ by continuity.)