Real and complex scalar conventions for normed spaces

The published definition A norm on a real vector space, the induced metric, and the dictionary with the metric axioms is written for vector spaces over $\mathbb{R}$. On this page the same language is used over $\mathbb{C}$ as well: a complex normed space is a complex vector space with a function $\mathrm{\parallel }\cdot \mathrm{\parallel }:V\to \mathbb{R}$ satisfying the same separation and triangle inequality clauses, while absolute homogeneity is read with the complex modulus $\mathrm{\mid }\lambda \mathrm{\mid }$ of Real and imaginary parts, complex conjugation, and modulus instead of the real absolute value.

Nothing else changes. The induced metric is still $d(x,y)=\mathrm{\parallel }x-y\mathrm{\parallel }$, Banach means complete for that metric, and every estimate on this page that uses only the triangle inequality and $\mathrm{\mid }\lambda \mathrm{\mid }$ is valid verbatim over either scalar field. When scalar continuity in the complex case is mentioned, the scalar field is the field of complex numbers already constructed in $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$.

• The page statements are therefore written so that the real case is literal and the complex case is obtained by this one substitution.
• Later pages that need genuinely complex-specific structure, such as sesquilinear inner products or adjoints, will say so explicitly rather than hiding it inside the word "normed".