3.I.1A

Define what is meant by a norm on a real vector space.

(a) Prove that two norms on a vector space (not necessarily finite-dimensional) give rise to equivalent metrics if and only if they are Lipschitz equivalent.

(b) Prove that if the vector space $V$ has an inner product, then for all $x, y \in V$,

$\|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2},$

in the induced norm.

Hence show that the norm on $\mathbb{R}^{2}$ defined by $\|x\|=\max \left(\left|x_{1}\right|,\left|x_{2}\right|\right)$, where $x=\left(x_{1}, x_{2}\right) \in$ $\mathbb{R}^{2}$, cannot be induced by an inner product.

*Typos? Please submit corrections to this page on GitHub.*