Exercise 12, § 2

Let E be an infinite dimensional real hilbertian space. Show that there exist infinitely many complex hilbertian space structures on E for which E is the real locally convex space underlying these complex hilbertian spaces (II, p. 61). (To prove the existence of the automorphisms u of the topological vector space structure of E, such that $u^2(x) = -x$, use an orthonormal basis of E; then apply V, p. 60, exerc. 1.) Give an example to show that the proposition does not extend to Hausdorff non complete prehilbertian spaces (consider an everywhere dense hyperplane in such a space).