Inlinear algebra,functional analysis and related areas ofmathematics, aquasinorm is similar to anorm in that it satisfies the norm axioms, except that thetriangle inequality is replaced by$${\displaystyle \Vert x+y\Vert \le K(\Vert x\Vert +\Vert y\Vert )}$$for some${\displaystyle K>1.}$

Aquasi-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:${\displaystyle p\ge 0;}$
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${\displaystyle k\ge 1}$ such that${\displaystyle p(x+y)\le k[p(x)+p(y)]}$ for all${\displaystyle x,y\in X.}$
   • If${\displaystyle k=1}$ then this inequality reduces to thetriangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.

Aquasinorm^[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 avector space${\displaystyle X}$ and an associated quasi-seminorm${\displaystyle p}$ is called aquasi-seminormed vector space. If the quasi-seminorm is a quasinorm then it is also called aquasinormed vector space.

Multiplier

Theinfimum of all values of${\displaystyle k}$ that satisfy condition (3) is called themultiplier 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${\displaystyle k.}$

Anorm (respectively, aseminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is${\displaystyle 1.}$ Thus everyseminorm is a quasi-seminorm and everynorm is a quasinorm (and a quasi-seminorm).

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]as${\displaystyle n}$ ranges over the positive integers. Atopological vector space with such a topology is called aquasinormed topological vector space or just aquasinormed space.

Every quasinormed topological vector space ispseudometrizable.

Acomplete quasinormed space is called aquasi-Banach space. EveryBanach space is a quasi-Banach space, although not conversely.

A quasinormed space${\displaystyle (A,\Vert \cdot \Vert )}$ is called aquasinormed algebra if the vector space${\displaystyle A}$ is analgebra and there is a constant${\displaystyle K>0}$ such that$${\displaystyle \Vert xy\Vert \le K\Vert x\Vert \cdot \Vert y\Vert }$$for all${\displaystyle x,y\in A.}$

Acomplete quasinormed algebra is called aquasi-Banach algebra.

Atopological 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, everynormed space is also a quasinormed space.

${\displaystyle L^p}$ spaces with

The${\displaystyle L^p}$ spaces for are quasinormed spaces (indeed, they are evenF-spaces) but they are not, in general,normable (meaning that there might not exist any norm that defines their topology). For theLebesgue space${\displaystyle L^p([0,1])}$ is acompletemetrizable TVS (anF-space) that isnotlocally convex (in fact, its onlyconvex open subsets are itself${\displaystyle L^p([0,1])}$ and the empty set) and theonlycontinuous linear functional on${\displaystyle L^p([0,1])}$ is the constant${\displaystyle 0}$ function (Rudin 1991, §1.47). In particular, theHahn-Banach theorem doesnot hold for${\displaystyle L^p([0,1])}$ when

• Metrizable topological vector space – Topological vector space whose topology can be defined by a metric
• Norm (mathematics) – Length in a vector space
• Seminorm – Mathematical function
• Topological vector space – Vector space with a notion of nearness

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• Linear algebra
• Norms (mathematics)