In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by
A quasi-seminorm[1] on a vector space
that satisfies the following conditions: A quasinorm[1] is a quasi-seminorm that also satisfies: A pair
consisting of a vector space
is called a quasi-seminormed vector space.
If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space.
Multiplier The infimum of all values of
that satisfy condition (3) is called the multiplier of
The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.
-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to
A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is
induces a vector topology on
whose neighborhood basis at the origin is given by the sets:[2]
ranges over the positive integers.
A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space.
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a quasi-Banach space.
is called a quasinormed algebra if the vector space
A complete quasinormed algebra is called a quasi-Banach algebra.
A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.
[2] Since every norm is a quasinorm, every normed space is also a quasinormed space.
are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology).
is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself
and the empty set) and the only continuous linear functional on
In particular, the Hahn-Banach theorem does not hold for