Seminorm

Let $X$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$. A function $p:X\to\mathbb{R}$ is called a seminorm if:

1. (Subadditivity) $p(x+y)\le p(x)+p(y)$ for all $x,y\in X$.
2. (Absolute homogeneity) $p(\lambda x)=|\lambda|\,p(x)$ for all $x\in X$ and all scalars $\lambda$.

Every norm is a seminorm, but a seminorm may vanish on nonzero vectors (e.g., $p(x_1,x_2)=|x_1|$ on $\mathbb{R}^2$).

Seminorms are the natural domination bounds in the seminorm versions of Hahn–Banach, including the real seminorm Hahn–Banach theorem and the complex seminorm Hahn–Banach theorem .