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All norms on a vector space $V$ must satisfy for any $x\in V$

$$\Vert \alpha x \Vert = \vert \alpha \vert \Vert x \Vert $$ for any scalar $\alpha\in R$.

However, I've been told that an equivalent condition is $$\Vert \alpha x \Vert \leq \vert \alpha \vert \Vert x \Vert .$$

Is this true, or is there a counterexample?

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1 Answer 1

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The relaxed condition also implies $$ \left\|\frac1\alpha \alpha x\right\|\le\left|\frac1\alpha\right|\|\alpha x\|$$ and hence $$ \|\alpha x\|\le |\alpha|\|x\|\le |\alpha|\left|\frac1\alpha\right|\|\alpha x\|=\|\alpha x\|,$$ which implies equality throughout.

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