Quasinorm

Definition:[1] A quasinorm on a vector space X is a real-valued map p on X that satisfies the following conditions:

1. Non-negativity: p ≥ 0;
2. Absolute homogeneity: p(sx) = |s| p(x) for all x ∈ X and all scalars s;
3. there exists a k ≥ 1 such that p(x + y) ≤ k[p(x) + p(y)] for all x, y ∈ 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:[1]

{ x ∈ X : p(x) < 1/n }

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 ${\displaystyle (A,\|\cdot \|)}$ is called a quasinormed algebra if the vector space A is an algebra and there is a constant K > 0 such that

${\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}$

for all ${\displaystyle x,y\in A}$.

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

A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.[1]

• Metrizable TVS
• Seminorm
• Topological vector space – Vector space with a notion of nearness

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