modular mappings in vector spaces over the field of complex numbers

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:

1. 1.

$\rho(x)=0$ if and only if $x=0$.

2. 2.

$\rho(\alpha x)=\rho(x)$ for all $x\in X$ and for all scalars $\alpha$ such that $|\alpha|=1$.

3. 3.

$\rho(\alpha x+\beta y)\leq\rho(x)+\rho(y)$ for all $x,y\in X$ and for all scalars $\alpha$ and $\beta$ such that $\alpha+\beta=1$ and $\alpha,\beta\geq 0$.

Title modular mappings in vector spaces over the field of complex numbers ModularMappingsInVectorSpacesOverTheFieldOfComplexNumbers 2013-03-22 16:08:12 2013-03-22 16:08:12 gilbert_51126 (14238) gilbert_51126 (14238) 11 gilbert_51126 (14238) Definition msc 46-00 modular