# 4.II.10E

(a) Let $V$ be a finite-dimensional real vector space, and let $\left\|_{1}\right\|_{1}$ and $\|\cdot\|_{2}$ be two norms on $V$. Show that a function $f: V \rightarrow \mathbb{R}$ is differentiable at a point $a$ in $V$ with respect to $\|\cdot\|_{1}$ if and only if it is differentiable at $a$ with respect to $\|\cdot\|_{2}$, and that if this is so then the derivative $f^{\prime}(a)$ of $f$ is independent of the norm used. [You may assume that all norms on a finite-dimensional vector space are equivalent.]

(b) Let $V_{1}, V_{2}$ and $V_{3}$ be finite-dimensional normed real vector spaces with $V_{j}$ having norm $\|\cdot\|_{j}, j=1,2,3$, and let $f: V_{1} \times V_{2} \rightarrow V_{3}$ be a continuous bilinear mapping. Show that $f$ is differentiable at any point $(a, b)$ in $V_{1} \times V_{2}$, and that $f^{\prime}(a, b)(h, k)=$ $f(h, b)+f(a, k) . \quad$ [You may assume that $\left(\|u\|_{1}^{2}+\|v\|_{2}^{2}\right)^{1 / 2}$ is a norm on $V_{1} \times V_{2}$, and that $\left\{(x, y) \in V_{1} \times V_{2}:\|x\|_{1}=1,\|y\|_{2}=1\right\}$ is compact.]