A quasi-seminorm^[1] on a vector space ${\displaystyle X}$  is a real-valued map ${\displaystyle p}$  on ${\displaystyle X}$  that satisfies the following conditions:

1. Non-negativity: ${\displaystyle p\geq 0;}$ 
2. Absolute homogeneity: ${\displaystyle p(sx)=|s|p(x)}$  for all ${\displaystyle x\in X}$  and all scalars ${\displaystyle s;}$ 
3. there exists a real ${\displaystyle k\geq 1}$  such that ${\displaystyle p(x+y)\leq k[p(x)+p(y)]}$  for all ${\displaystyle x,y\in X.}$ 
   • If ${\displaystyle k=1}$  then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.

A quasinorm^[1] is a quasi-seminorm that also satisfies:

4. Positive definite/Point-separating: if ${\displaystyle x\in X}$  satisfies ${\displaystyle p(x)=0,}$  then ${\displaystyle x=0.}$ 

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

Multiplier

The infimum of all values of ${\displaystyle k}$  that satisfy condition (3) is called the multiplier of ${\displaystyle p.}$  The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term ${\displaystyle k}$ -quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to ${\displaystyle k.}$ 

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

Topology edit

If ${\displaystyle p}$  is a quasinorm on ${\displaystyle X}$  then ${\displaystyle p}$  induces a vector topology on ${\displaystyle X}$  whose neighborhood basis at the origin is given by the sets:^[2]

Every quasinormed topological vector space is pseudometrizable.

A complete quasinormed space is called a quasi-Banach space. Every Banach space is a quasi-Banach space, although not conversely.

Related definitions edit

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

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.^[2]

Since every norm is a quasinorm, every normed space is also a quasinormed space.

${\displaystyle L^{p}}$  spaces with ${\displaystyle 0<p<1}$ 

The ${\displaystyle L^{p}}$  spaces for ${\displaystyle 0<p<1}$  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 ${\displaystyle 0<p<1,}$  the Lebesgue space ${\displaystyle L^{p}([0,1])}$  is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself ${\displaystyle L^{p}([0,1])}$  and the empty set) and the only continuous linear functional on ${\displaystyle L^{p}([0,1])}$  is the constant ${\displaystyle 0}$  function (Rudin 1991, §1.47). In particular, the Hahn-Banach theorem does not hold for ${\displaystyle L^{p}([0,1])}$  when ${\displaystyle 0<p<1.}$ 

• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Norm (mathematics) – Length in a vector space
• Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback
• Topological vector space – Vector space with a notion of nearness

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