modular mappings in vector spaces over the field of complex numbers
(Definition)
Suppose $X$ is a $\mathbb{C}$ -vector space. A mapping$\rho:X\to [0,\infty]$ is called modular if the following three conditions are satisfied:
$\rho(x) = 0$ if and only if $x=0$ .
$\rho(\alpha x) = \rho(x)$ for all $x \in X$ and for all scalars$\alpha$ such that $|\alpha|=1$ .
$\rho(\alpha x + \beta y) \leq \rho(x) + \rho(y)$ for all $x,y \in X$ and for all scalars $\alpha$ and $\beta$ such that $\alpha + \beta =1$ and $\alpha ,\beta \geq 0$ .
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