Exercise 2, § 1

Let E be a real or complex normed space. Suppose that for every 2-dimensional vector subspace P (over $\mathbf{R}$) in E, there exists a symmetric bilinear form $f_P$ defined on $P \times P$, such that $f_P(x, x) = \|x\|^2$ for all $x \in P$. Show that $f_P$ is defined unambiguously and that there exists a hermitian sesquilinear form $g$ on $E \times E$ such that, for every real plane $P \subset E$, we have $f_P(x, y) = \Re g(x, y)$, hence $g(x, x) = \|x\|^2$. (If E is a real vector space, observe that we have $\|x - y\|^2 + \|x + y\|^2 = 2(\|x\|^2 + \|y\|^2)$ for every pair of points of E, and deduce the identity

if E is a complex vector space, apply exerc. 1.)