What is meant by a norm on $\mathbb{R}^{n}$ ? For $\mathbf{x} \in \mathbb{R}^{n}$ write

$$\begin{gathered} \|\mathbf{x}\|_{1}=\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right| \\ \|\mathbf{x}\|_{2}=\sqrt{\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\cdots+\left|x_{n}\right|^{2}} \end{gathered}$$

Prove that $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ are norms. [You may assume the Cauchy-Schwarz inequality.]

Find the smallest constant $C_{n}$ such that $\|x\|_{1} \leqslant C_{n}\|x\|_{2}$ for all $x \in \mathbb{R}^{n}$, and also the smallest constant $C_{n}^{\prime}$ such that $\|x\|_{2} \leqslant C_{n}^{\prime}\|x\|_{1}$ for all $x \in \mathbb{R}^{n}$.