(a) What is a norm on a real vector space?

(b) Let $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$ be the space of linear maps from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$. Show that

$$\|A\|=\sup _{0 \neq x \in \mathbb{R}^{m}} \frac{\|A x\|}{\|x\|}, \quad A \in L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right),$$

defines a norm on $L\left(\mathbb{R}^{m}, \mathbb{R}^{n}\right)$, and that if $B \in L\left(\mathbb{R}^{\ell}, \mathbb{R}^{m}\right)$ then $\|A B\| \leqslant\|A\|\|B\|$.

(c) Let $M_{n}$ be the space of $n \times n$ real matrices, identified with $L\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)$ in the usual way. Let $U \subset M_{n}$ be the subset

$$U=\left\{X \in M_{n} \mid I-X \text { is invertible }\right\}$$

Show that $U$ is an open subset of $M_{n}$ which contains the set $V=\left\{X \in M_{n} \mid\|X\|<1\right\}$.

(d) Let $f: U \rightarrow M_{n}$ be the map $f(X)=(I-X)^{-1}$. Show carefully that the series $\sum_{k=0}^{\infty} X^{k}$ converges on $V$ to $f(X)$. Hence or otherwise, show that $f$ is twice differentiable at 0 , and compute its first and second derivatives there.