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Let $(E, \| \|)$ be a normed space.

We define $\phi:E \rightarrow [0,\infty)$ as follows:

$$\phi(e)= \dfrac{\|e\|}{1+\|e\|}$$

Is $\phi$ a norm of $E$?

Please help! Thank you!

P.S. This question looks familiar to Show that d_b is metric, but I think it asks different thing.

Community
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John Lennon
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1 Answers1

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No it is not a norm. If $\|x\|\ne 0$, then $$ \varphi(2x)=\frac{\|2x\|}{1+\|2x\|}=\frac{2\|x\|}{1+2\|x\|}\ne 2\frac{\|x\|}{1+\|x\|}=2\varphi(x). $$

Yiorgos S. Smyrlis
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  • thank you! In these kinds of questions we basically just take one of the norm properties and find a counterexample,right? – John Lennon Oct 27 '14 at 07:48