Question

Show that a discrete metric on a vector space X≠{0} cannot be obtained from a norm (the pair (X,d) is called the discrete metric space if the metric is d(x,x)=0, d(x,y)=1 for x≠y).

Answer 1

You know that the discrete metric only takes values of 1 and 0. Now suppose it comes from some norm ||.||. Then for any α in the underlying field of your vector space and x,y∈X, you must have that

∥α(x−y)∥=|α|∥x−y∥.

But now ||x−y|| is a fixed number and I can make α arbitrarily large and consequently the discrete metric does not come from any norm on X.

Explanation:

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