Metrics induced by norms and inner products

Let $X$ be a linear metric space with a metric $\rho$ having the following properties:

(i) $\forall x, y, z \in X: \rho (x+z, y+z) = \rho (x, y)$;

(ii) $\forall x \in X \forall \lambda \in \mathbb {C}: \rho (0, \lambda x) = \left|\lambda \right| \rho (0, x)$.

Letting $A$, $B$, and $C$ represent arbitrary matrices in a linear space $\mathcal{V}$, show that

A normed space $X$ is a metric space with the metric $\rho (x ; y) = \left\| x-y\right\|_{X}$.

Give an example of a metric space in which the metric is not generated by a norm.

In the linear space of all numerical sequences, is the metric of problem 18.14 generated by some norm?

Prove Theorem 1.1.

Theorem 1.1: Norms always give rise to metrics. Specifically, if $(V, \left\| \cdot \right\| )$ is a normed space and $d(v, w)$ is defined by $d(v, w) = \left\| v - w\right\|$, then $d$ is a metric on $V$.

Prove Theorem 1.2. (Hint: Use the Cauchy-Schwarz inequality to get the triangle inequality.)

Theorem 1.2: Inner products always give rise to norms. Specifically, if $(V, \left\langle \cdot , \cdot \right\rangle )$ is an inner product space and $\left\| v\right\|$ is defined by $\left\| v\right\| = \sqrt{ \left\langle v, v \right\rangle }$, then $\left\| \cdot \right\|$ is a norm on $V$.

Prove that in any complex inner product space

for every $v, w \in V$ and each $\lambda \in \mathbb {C}$.

Consider a normed linear space $(V,\left\| \cdot \right\| )$. Recall that some normed linear spaces are inner product spaces, and some are not.

Show that the metrics $d_{1}$ and $d_{2}$ from item 5 are not induced by any inner product on $\mathbb {R}^{n}$ (similarly on $\mathbb {C}^{n}$).