I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to do it in 15.)
First part: prove that two norms on a vector space (not necessarily finite-dimensional) give rise to equivalent metrics iff they are Lipschitz equivalent.
I did this by assuming the metrics are equivalent and showing that any $d$-open ball is contained within a d'-open ball and thence that the norms are Lipschitz equivalent; and conversely, if they are Lipschitz equivalent, then any $d$-open ball is contained within a d'-open ball.
Second part: prove that if the vector space V has an inner product, then for all $x,y\in V$, $\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2$.
I did this by using the definition of the norm in terms of the inner product and cancelling $
Third part: hence show that the norm on $\mathbb{R}^2$ defined by $\|x\|=\max{(|x_1|,|x_2|)}$, where $x=(x_1,x_2)\in\mathbb{R}^2$, cannot be induced by an inner product.
I did this by using $x=\begin{pmatrix} 1\\0 \end{pmatrix}$ and $y=\begin{pmatrix} 0\\1 \end{pmatrix}$ as the counterexample for the uniform norm.
What am I missing? How could I have done any of these parts quicker?