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I have a question with an exercise about norm. Let $C_{\left[0,1\right]}$ is a continous funtion space on $\left[0,1\right]$. A funtion $\|{\cdot}\|:C_{\left[0,1\right]} \rightarrow \mathbb{R}$ is definded by $\|x\|=\max_{t\in\left[0,1\right]}|x(t)|^\alpha$. Find $\alpha$ for $\|{\cdot}\|$ be a norm.

With the first two propositions of definition of norm, it's easy to show that $\alpha\neq 0$ cause in that case $\|0\|=|0|^0=1$, but i have a problem with the last proposition. So which $\alpha$ have the property $\|x+y\|^\alpha\leq \|x\|^\alpha+\|y\|^\alpha$?

I proved it's right with $\alpha\in(0,1]$ , but the other I can't. Thank you for answering.

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Define $x \in C[0,1]$ by $x(t))=1$ for $t \in [0,1]$. Then $||x||=1$. If $||*||$ is a norm, then $||2x||=2 ||x||=2$. On the other hand we have $||2x||=2^{ \alpha}$. Thus

$$2=2^{\alpha}.$$

Conclusion ?

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  • $\begingroup$ oh i didn't think in that way, thank for your answer $\endgroup$ – Hoàng Nov 2 '18 at 11:45

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