[kanji]

1.Prove that for $x,y\in R^k$
(a.) $\left\Vert\,x\right\Vert\not= 0$ with equality only when $x=0$.
(b.) $\left\Vert\,\alpha x\right\Vert=\left|\,\alpha \right| \left\Vert\,x\right\Vert$ for all scalars $\alpha$.
(c.) $\left\Vert\,x+y\right\Vert\leq \left\Vert\,x\right\Vert+\left\Vert\,y\right\Vert$ and
$\left\Vert\,x-y\right\Vert\geq \left\Vert\,x\right\Vert-\left\Vert\,y\right\Vert$.
(d.) $\left\Vert\,x\right\Vert\leq \sum_{i = 1}^{k} \left\Vert\,x_i\right\Vert$.
(e.) $\left|\,x_i \right|\leq \left\Vert\,x\right\Vert\leq \sqrt{n}\left\Vert\,x_\infty \right\Vert$ for each $i=1,2,3,...,k$.