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A non-negative function defined on a linear space and satisfying the same axioms as a norm except for the triangle inequality , which is replaced by the weaker requirement: There exists a constant such that for all .


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 in a topological vector space is bounded if for each open neighbourhood of zero there is a such that , and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood of zero in a topological vector space (a set is circled if for all ), the Minkowski functional of is defined by . It is a quasi-norm.


[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 159
How to Cite This Entry:
Quasi-norm. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article