Let $\mathbb{C}^N$ be a D5640: Set of euclidean complex numbers.
Let $|\cdot| : \mathbb{C} \to [0, \infty)$ be the D1210: Complex modulus function.
The complex euclidean P-length function on $\mathbb{C}^N$ with respect to $p \in (0, \infty)$ is the D4365: Unsigned Realll func function \begin{equation} \Vert \cdot \Vert_p : \mathbb{C}^N \to [0, \infty), \quad \Vert z \Vert_p = \left( \sum_{n = 1}^N |z_n|^p \right)^{1 / p} \end{equation}