# Quasi-norm

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A non-negative function $\|x\|$ defined on a linear space $R$ and satisfying the same axioms as a norm except for the triangle inequality $\|x+y\|\leq\|x\|+\|y\|$, which is replaced by the weaker requirement: There exists a constant $c>0$ such that $\|x+y\|\leq c(\|x\|+\|y\|)$ for all $x,y\in R$.
The topology of a locally bounded topological vector space can be given by a quasi-norm. Conversely, a quasi-normed vector space is locally bounded. Here a set $B$ in a topological vector space is bounded if for each open neighbourhood $U$ of zero there is a $\rho>0$ such that $B\subset\rho U$, and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood $U$ of zero in a topological vector space $E$ (a set $M\subset E$ is circled if $\alpha M\subset M$ for all $|\alpha|\leq1$), the Minkowski functional of $U$ is defined by $q(x)=\inf_{x\in\alpha U}\alpha$. It is a quasi-norm.