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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 $<x,y>$ terms.

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?

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This looks like the way to go in all three problems. $7.5$ minutes is a bit tight to write out the details, I'd say. – t.b. Apr 4 '12 at 21:38
Why these time limits?? – lhf Apr 4 '12 at 21:54
The 7.5 minutes is how long I'd have for the question if all marks took the same amount of time to get. I'm doing lots of past paper questions in preparation for an exam. Some take less and some take a few minutes more, but 15 minutes makes this question (or rather, my answer to it!) an outlier. Just wondered whether I was missing something that would enable it to be done significantly quicker, but maybe the answer is no? – Dave Carruthers Apr 4 '12 at 22:08

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