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.


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


Conclusion ?

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.