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

Definition
A on a vector space $$X$$ is a real-valued map $$p$$ on $$X$$ that satisfies the following conditions:  : $$p \geq 0;$$ : $$p(s x) = |s| p(x)$$ for all $$x \in X$$ and all scalars $$s;$$ there exists a real $$k \geq 1$$ such that $$p(x + y) \leq k [p(x) + p(y)]$$ 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: 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: $$\{x \in X : p(x) < 1/n\}$$ 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 $$L^p([0, 1])$$ is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself $$L^p([0, 1])$$ and the empty set) and the  continuous linear functional on $$L^p([0, 1])$$ is the constant $$0$$ function. In particular, the Hahn-Banach theorem does hold for $$L^p([0, 1])$$ when $$0 < p < 1.$$