Quasi-norm

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
$\, x + y\, \leq K(\, x\, + \, y\, )$
for some $K > 1.$

Definition

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

1. : $p \geq 0;$


2. : $p(s x) = , s, p(x)$ for all $x \in X$ and all scalars $s;$


3. there exists a real $k \geq 1$ such that p(x + y) \leq k (x) + p(y)/math> for all $x, y \in X.$
   * If $k = 1$ 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:

4. Positive definite/: if $x \in X$ satisfies $p(x) = 0,$ then $x = 0.$

A pair $(X, p)$ consisting of a vector space $X$ and an associated quasi-seminorm $p$ is called a .
If the quasi-seminorm is a quasinorm then it is also called a .

Multiplier

The infimum of all values of $k$ that satisfy condition (3) is called the of $p.$
The multiplier itself will also satisfy condition (3) 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 $k.$

A (respectively, a ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is $1.$
Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).

Topology

If $p$ is a quasinorm on $X$ then $p$ induces a vector topology on $X$ whose neighborhood basis at the origin is given by the sets:
$\$
as $n$ 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 $(A, \, \,\cdot\, \, )$ is called a if the vector space $A$ is an algebra and there is a constant $K > 0$ such that
$\, x y\, \leq K \, x\, \cdot \, y\,$
for all $x, y \in A.$

A complete quasinormed algebra is called a .

Characterizations

A topological vector space (TVS) 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.

$L^p$ spaces with $0 < p < 1$

The $L^p$ spaces for $0 < p < 1$ 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 $0 < p < 1,$ the Lebesgue space $L^p($, 1 is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself $L^p($, 1 and the empty set) and the continuous linear functional on $L^p($, 1 is the constant $0$ function .
In particular, the Hahn-Banach theorem does hold for $L^p($, 1 when $0 < p < 1.$

See also

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References

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{{Banach spaces

Linear algebra
Norms (mathematics)