Quasinorm

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

Definition

A on a vector space is a real-valued map on that satisfies the following conditions:

:
: for all and all scalars
there exists a real such that for all
- If then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.

A is a quasi-seminorm that also satisfies:

Positive definite/: if satisfies then

A pair consisting of a vector space and an associated quasi-seminorm is called a.
If the quasi-seminorm is a quasinorm then it is also called a.
Multiplier
The infimum of all values of that satisfy condition is called the of
The multiplier itself will also satisfy condition and so it is the unique smallest real number that satisfies this condition.
The term is sometimes used to describe a quasi-seminorm whose multiplier is equal to
A is just a quasinorm whose multiplier is
Thus every seminorm is a quasi-seminorm and every norm is a quasinorm.

Topology

If is a quasinorm on then induces a vector topology on whose neighborhood basis at the origin is given by the sets:
as ranges over the positive integers.
A topological [vector space] with such a topology is called a or just a.
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a. Every Banach space is a quasi-Banach space, although not conversely.

Related definitions

A quasinormed space is called a if the vector space is an algebra and there is a constant such that
for all
A complete quasinormed algebra is called a.

Characterizations

A topological vector space is a quasinormed space if and only if it has a bounded neighborhood of the origin.

Examples

Since every norm is a quasinorm, every normed space is also a quasinormed space.
spaces with
The spaces for are quasinormed spaces but they are not, in general, normable.
For the Lebesgue space is a complete metrizable TVS that is locally convex and the continuous linear functional on is the constant function.
In particular, the Hahn-Banach theorem does hold for when