# Find $\alpha$ for $\|{\cdot}\|$ be a norm

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