Chapter 10.3, Problem 34E

Norm. The norm of a function $\Vert f\Vert$ is like the length of a vector in $R^n$. In particular, show that the norm defined in $\left(16\right)$ satisfies the following properties associated with length (assume $f$ and $g$ are continuous and $w\left(x\right)>0$ on [a, b]):

a. $\Vert f\Vert \ge 0,$ and $\Vert f\Vert =0$ if and only if $f\equiv 0$.

b. $\Vert cf\Vert =\left|c\right|\Vert f\Vert$, where $c$ is any real number.

c. $\Vert f+g\Vert \le \Vert f\Vert +\Vert g\Vert .$

d.