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
for some
A quasi-seminorm [1] on a vector space is a real-valued map on that satisfies the following conditions:
A quasinorm [1] is a quasi-seminorm that also satisfies:
A pair consisting of a vector space and an associated quasi-seminorm 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. The term -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 Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
If is a quasinorm on then induces a vector topology on whose neighborhood basis at the origin is given by the sets: [2]
as 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. Every Banach space is a quasi-Banach space, although not conversely.
A quasinormed space is called a quasinormed algebra if the vector space is an algebra and there is a constant such that
for all
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.
spaces with
The spaces for 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). For the Lebesgue space 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 is the constant function ( Rudin 1991 , §1.47). In particular, the Hahn-Banach theorem does not hold for when
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