Quasi-Banach Space

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 $K > 0.$

Related concepts

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

1. Non-negativity: $p \geq 0;$


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


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

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 (TVS) with such a topology is called a quasinormed space.

Every quasinormed TVS is a pseudometrizable.

A vector space with an associated quasinorm is called a quasinormed vector space.

A complete quasinormed space is called a quasi-Banach space.

A quasinormed space $(A, \, \,\cdot\, \, )$ is called a quasinormed algebra 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 quasi-Banach algebra.

Characterizations

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

See also

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References

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{{Topological vector spaces

Linear algebra
Norms (mathematics)