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Let $$\|f\|_{\max}=\max_{x\in[0,1]}|f(x)|$$

Is $\|f\|_{\max}$ a norm?

When looking at homogeneity we start with $\lambda\in \mathbb{F}$ and $f\in C[0,1]$:

$$\|\lambda\cdot f\|_{\max}=\max_{x\in[0,1]}|\lambda\cdot f(x)|=|\lambda|\cdot \max_{x\in[0,1]}| f(x)|=|\lambda| \|f\|_{\max}$$

it is intuitive that we can "pull out" the $\lambda$ as the max function gets value of the image of a given function and does not "work" with scalars.

But how can we say it in a formal way?

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If $c \ge 0$ then $\sup_x c h(x) = c \sup_x h(x)$.

This is obvious if $c=0$, so suppose $c>0$. Then note that $h(x) \le \sup_x h(x)$, so $c h(x) \le c \sup_x h(x)$ and hence $\sup_x c h(x) \le c \sup_x h(x)$. To obtain the other direction, note that the last result shows that $\sup_x h(x) \le {1 \over c} \sup_x c h(x)$.

If $|f(x)+g(x)| \le |f(x)|+ |g(x)|$, then (with slight abuse, or overuse, of notation) $|f(x)+g(x)| \le \sup_x |f(x)| + \sup_x |g(x)|$ from which it follows that $\sup_x |f(x)+g(x)| \le \sup_x |f(x)| + \sup_x |g(x)|$.

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