Exercise 24, § 7

Let $E$ be a vector space of finite dimension $n > 1$ over a commutative field $K$; show that, unless $n = 2$ and $K$ is a field with two elements, there exists no isomorphism $\phi$ of $E$ onto $E^*$ depending only on the vector space structure of $E$. (Note that if $\phi$ is such an isomorphism, then necessarily $\langle x, \phi(y) \rangle = \langle u(x), \phi(u(y)) \rangle$ for $x, y$ in $E$ and for every automorphism $u$ of $E$ (Set Theory, IV, § 1, no. 5).)