(d) Does $\langle\cdot, \cdot\rangle$ defined on $\mathbb{R} \times \mathbb{R}$ by $\langle x, y\rangle=x y$ define an inner product on $\mathbb{R}$? What about $\langle z, w\rangle=z \bar{w}$ on $\mathbb{C}$?
(e) Let $\langle\cdot, \cdot\rangle$ be an inner product. Show that if $\langle x, y\rangle=\langle x, z\rangle$ for all $x$; then $y=z$.
