Let $V$ be a \reftext{def:vector-space-2026a}{complex vector space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, and for a \reftext{def:complex-numbers-2026a}{complex number} $\lambda$ let $|\lambda|$ denote its \reftext{def:complex-modulus-2026a}{modulus}.

A \textbf{norm} on $V$ is a map assigning to each $v\in V$ a \reftext{def:real-numbers-c54-2026c}{real number} $\lVert v\rVert$, subject to the following conditions for all $u,v\in V$ and all complex $\lambda$.

\textbf{1. (Positivity)} $0\le\lVert v\rVert$, and $\lVert v\rVert=0$ only if $v=0_{V}$.

\textbf{2. (Absolute homogeneity)} $\lVert\lambda v\rVert=|\lambda|\,\lVert v\rVert$.

\textbf{3. (Triangle inequality)} $\lVert u+v\rVert\le\lVert u\rVert+\lVert v\rVert$.

A \textbf{complex normed space} is a complex vector space together with a norm on it.