1. Non-negativity: \(\|x\| \ge 0\).
2. Zero only at the zero vector: \(\|x\| = 0\) exactly when \(x=0\).
3. Homogeneity: \(\|a x\| = \lvert a\rvert \,\|x\|\) for any scalar \(a\).
4. Triangle inequality: \(\|x+y\| \le \|x\| + \|y\|\).