Exercise 1, § 3

Let E be a hilbertian space. Show that the bilinear mapping $(u, v) \mapsto uv$ from $\hat{\mathbf{S}}^m(E) \times \hat{\mathbf{S}}^n(E)$ into $\hat{\mathbf{S}}^{m+n}(E)$ is continuous and that its norm is equal to $\left( \frac{(m+n)!}{m! \ n!} \right)^{1/2}$. (To see that this norm is bounded by $\left( \frac{(m+n)!}{m! \ n!} \right)^{1/2}$, argue as in the case of the exterior algebra (V, p. 35). Deduce that the multiplication in $\mathbf{S}(E)$ cannot be extended to $\hat{\mathbf{S}}(E)$ by continuity when E is not simply 0.