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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:

  1. $ \rho(x) = 0$ if and only if $ x=0$.
  2. $ \rho(\alpha x) = \rho(x)$ for all $ x \in X$ and for all scalars $ \alpha$ such that $ \vert\alpha\vert=1$.
  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$.



"modular mappings in vector spaces over the field of complex numbers" is owned by gilbert_51126. [ full author list (2) ]
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Also defines:  modular
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Cross-references: scalars, mapping
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This is version 8 of modular mappings in vector spaces over the field of complex numbers, born on 2006-08-03, modified 2006-09-18.
Object id is 8208, canonical name is ModularMappingsInVectorSpacesOverTheFeildOfComplexNumbers.
Accessed 1820 times total.

Classification:
AMS MSC: 46-00 (Functional analysis :: General reference works )
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