Solutions

a) We have

$$\begin{align*} \|cf\| &= \max_{x\in[0,1]} |cf(x)|\\ &= \max_{x\in[0,1]} |c|\,|f(x)|\\ &= |c|\max_{x\in[0,1]} |f(x)|\\ &= |c|\,\|f\|. \end{align*}$$

b) Consider $\alpha$ for which $\|f+g\|=|f(\alpha)+g(\alpha)|$. It follows by the triangle inequality that

$$\|f+g\|=|f(\alpha)+g(\alpha)|\le |f(\alpha)|+|g(\alpha)|\le \|f\|+\|g\|.$$

Notice that if $f(x)=g(x)\ne0$, then $\|f+g\|\ne\|f\|+\|g\|$.

c) The result follows from b) if $r=h-g$, $s=g-f$, and $\|r+s\|\le\|r\|+\|s\|$.