Prove that $N_2(\mathbf z)=(\sum_{k=1}^n \overline {z_k}z_k)^{1/2}$ is a norm on $\mathbb C^n$

Question

Here $\mathbf z=\{z_k=x_k+iy_k\}_{k=1}^n, \overline{\mathbf z}=\{x_k-iy_k\}_{k=1}^n \in \mathbb C^n$.

So far I have proven the following properties required for $N_2(\mathbf z)$ to be a norm:

• $\parallel \mathbf z \parallel \geq \mathbf 0$ for all $\mathbf z \in V$, where $ V$ is a complex vector space.

• If $\parallel \mathbf z \parallel=0$, then $\mathbf z=\mathbf 0$

• If $\lambda \in \mathbb C$ and $\mathbf z \in V$, then $\parallel\lambda \mathbf z\parallel=\mid\lambda\mid\cdot\parallel\mathbf z\parallel$

I am struggling to prove the last property though which is the triangle inequality, i.e If $\mathbf z, \mathbf w\in V$, then $\parallel\mathbf z+\mathbf w\parallel\leq\parallel \mathbf z\parallel + \parallel \mathbf w \parallel$. Can anyone help me to prove it?

Answer 1

It follows from Cauchy's inequality: \begin{align*}(\|z\|+\|w\|)^2&=\|z\|^2+\|w\|^2+2\|z\|\|w\|\ge\|z\|^2+\|w\|^2+2|\langle z,w\rangle|\ge\|z\|^2+\|w\|^2+2\text{Re}\,\langle z,w\rangle\\&=\|z\|^2+\|w\|^2+\langle z,w\rangle+\overline{\vphantom{x}\smash{\langle z,w\rangle}}=\langle z,z\rangle+\langle z,w\rangle+\langle w,z\rangle+\langle w,w\rangle=\langle z+w,z+w\rangle\\&=\|z+w\|^2\end{align*}