Quasinorm

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

Definition

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

Non-negativity: $p\geq 0;$
Absolute homogeneity: $p(sx)=|s|p(x)$ for all $x\in X$ and all scalars $s;$
there exists a real $k\geq 1$ $k\geq 1$ such that $p(x+y)\leq k[p(x)+p(y)]$ $p(x+y)\leq k[p(x)+p(y)]$ for all $x,y\in X.$ $x,y\in X.$
- If $k=1$ 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:

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

A pair $(X,p)$ consisting of a vector space $X$ and an associated quasi-seminorm $p$ 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 $k$ that satisfy condition (3) is called the multiplier of $p.$ The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term $k$ -quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to $k.$

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

Topology

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:^[2] $\{x\in X:p(x)<1/n\}$ as $n$ 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 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.^[2]

Examples

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

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

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

References

^ ^a ^b Kalton 1986, pp. 297–324.
^ ^a ^b 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.

[FOOTNOTEKalton1986297–324-1] Kalton 1986, pp. 297–324.

[FOOTNOTEWilansky201355-2] Wilansky 2013, p. 55.

[1]

[2]

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics