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If ##d(x,y)## is a metric, then it is said ##\frac{d}{1+d}## is also a metric. I don't know the proof of this, I'd appreciate a reference, but it got me wondering:

If ##N(x)## is a norm on a Banach space ##x \in X##, then are there functions in a single real (or complex) variable ##f## (besides the identity function) for which ##f(N)## is also a norm? What would be the purpose of different norms of this nature?

If ##N(x)## is a norm on a Banach space ##x \in X##, then are there functions in a single real (or complex) variable ##f## (besides the identity function) for which ##f(N)## is also a norm? What would be the purpose of different norms of this nature?

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