Is $\phi$ a norm of E?

Question

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.

Answer 1

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