Neha Patil (Editor)

Quasinorm

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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 ∥ ≤ K ( ∥ x ∥ + ∥ y ∥ )

for some K > 0.

This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satisfied except for positive definiteness.

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

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

A quasinormed space ( A , ∥ ⋅ ∥ ) is called a quasinormed algebra if the vector space A is an algebra and there is a constant K > 0 such that

∥ x y ∥ ≤ K ∥ x ∥ ⋅ ∥ y ∥

for all x , y ∈ A .

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

References

Quasinorm Wikipedia
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