A pseudometric on a set $${\displaystyle X}$$ is a map $${\displaystyle d:X\times X\rightarrow \mathbb {R} }$$ satisfying the following properties:
1. See more

If $${\displaystyle X}$$ is a vector space over the real or complex numbers then a paranorm on $${\displaystyle X}$$ is a G-seminorm (defined … See more

Suppose that $${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$$ is a family of non-negative subadditive functions on a vector space $${\displaystyle X.}$$ See more

• Every seminormed space $${\displaystyle (X,p)}$$ is pseudometrizable with a canonical pseudometric given by $${\displaystyle d(x,y):=p(x-y)}$$ for all
• If See more

An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators. See more

If $${\displaystyle X}$$ is a vector space over the real or complex numbers then an F-seminorm on $${\displaystyle X}$$ (the $${\displaystyle F}$$ stands for Fréchet) is a real-valued map $${\displaystyle p:X\to \mathbb {R} }$$ with the following four properties: See more

Of (pseudo)metrics induced by (semi)norms
A pseudometric (resp. metric) $${\displaystyle d}$$ is induced by a seminorm (resp. norm) on a vector space $${\displaystyle X}$$ if and only if $${\displaystyle d}$$ See more