Exercise 1, § 1

Let E be a complex normed space and $f$ a symmetric bilinear form on the underlying real vector space $E_0$, such that $f(x, x) = \|x\|^2$ for all $x \in E$. Show that there exists one and only one hermitian sesquilinear form $g$ on E such that $f(x, y) = \Re g(x, y)$ (prove that $f(x, iy) = -f(ix, y)$ by using formula (5) of V, p. 2), hence $g(x, x) = \|x\|^2$.