Norm on a Complex Vector Space

Let $V$ be a complex vector space with zero vector $0_{V}$, and for a complex number $\lambda$ let $|\lambda|$ denote its modulus.

A norm on $V$ is a map assigning to each $v\in V$ a real number $\lVert v\rVert$, subject to the following conditions for all $u,v\in V$ and all complex $\lambda$.

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

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

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

A complex normed space is a complex vector space together with a norm on it.