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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 $${\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)}$$ for some ${\displaystyle K>1.}$

Definition

A quasi-seminorm 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 ${\displaystyle p(x+y)\leq k}$ 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 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

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: $${\displaystyle \{x\in X:p(x)<1/n\}}$$ as ${\displaystyle n}$ ranges over the positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space.

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

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 $${\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}$$ for all ${\displaystyle x,y\in A.}$

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.

Examples

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

See also

• Metrizable topological vector space – A 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

References

1. ^ Kalton 1986, pp. 297–324.
2. ^ Wilansky 2013, p. 55.

• Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. Springer. ISBN 0-7923-6970-X.
• Conway, John B. (1990). A Course in Functional Analysis. Springer. ISBN 0-387-97245-5.
• Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces" (PDF). Studia Mathematica. 84 (3). Institute of Mathematics, Polish Academy of Sciences: 297–324. doi:10.4064/sm-84-3-297-324. ISSN 0039-3223.
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. Vol. 19. Springer. ISBN 3-540-50584-9.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Swartz, Charles (1992). An Introduction to Functional Analysis. CRC Press. ISBN 0-8247-8643-2.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

v t e Lp spaces
Basic concepts	Banach & Hilbert spaces L spaces Measure Lebesgue Measure space Measurable space/function Minkowski distance Sequence spaces
L spaces	Integrable function Lebesgue integration Taxicab geometry
L spaces	Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function
${\displaystyle L^{\infty }}$ spaces	Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm
Maps	Almost everywhere Convergence almost everywhere Convergence in measure Function space Integral transform Locally integrable function Measurable function Symmetric decreasing rearrangement
Inequalities	Babenko–Beckner Chebyshev's Clarkson's Hanner's Hausdorff–Young Hölder's Markov's Minkowski Young's convolution
Results	Marcinkiewicz interpolation theorem Plancherel theorem Riemann–Lebesgue Riesz–Fischer theorem Riesz–Thorin theorem For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality
Applications & related	Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann

• Linear algebra
• Norms (mathematics)