I currently in the midst of a calculation and during this calculation I ran into the question as to whether or not there existed $p$-norms that included $p \in \mathbb{C}$. I know that a $p$-norm is defined as $$||\overrightarrow{x}||_p := \left(\sum_{k=1}^{n} |x_{k}|^p\right)^\frac{1}{p}$$ for $p \geq 1$. This is because only these values of $p$ satisfy the requirements for being a norm, in that:
- $||\overrightarrow{x}+\overrightarrow{y}||_p \leq ||\overrightarrow{x}||_p +||\overrightarrow{y} ||_p$ (Triangle Inequality)
- $||c \cdot \overrightarrow{x}||_p = |c| \cdot ||\overrightarrow{x}||_p$ (Absolutely Homogeneous)
- If $||\overrightarrow{x}||_p = 0$, then $\overrightarrow{x} = \overrightarrow{0}$ (Positive Definite)
Despite these criteria, there do exist certain looser "norms", such as semi-norms, the zero norm or $p$ norms with values in the range $(0, 1)$. I'm merely curious if such a "pseudo-norm" exists for complex values of $p$.