Inlinear algebra, asublinear function (orfunctional as is more often used infunctional analysis), also called aquasi-seminorm or aBanach functional, on avector space${\displaystyle X}$ is areal-valuedfunction with only some of the properties of aseminorm. Unlike seminorms, a sublinear function does not have to benonnegative-valued and also does not have to beabsolutely homogeneous. Seminorms are themselves abstractions of the more well known notion ofnorms, where a seminorm has all the defining properties of a normexcept that it is not required to map non-zero vectors to non-zero values.

Infunctional analysis the nameBanach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of theHahn–Banach theorem. The notion of a sublinear function was introduced byStefan Banach when he proved his version of theHahn-Banach theorem.^[1]

There is also a different notion incomputer science, described below, that also goes by the name "sublinear function."

Let${\displaystyle X}$ be avector space over a field${\displaystyle \mathbb{K},}$ where${\displaystyle \mathbb{K}}$ is either thereal numbers${\displaystyle \mathbb{R}}$ orcomplex numbers${\displaystyle \mathbb{C}.}$ A real-valued function${\displaystyle p:X\to \mathbb{R}}$ on${\displaystyle X}$ is called asublinear function (or asublinearfunctional if${\displaystyle \mathbb{K}=\mathbb{R}}$), and also sometimes called aquasi-seminorm or aBanach functional, if it has these two properties:^[1]

1. Positive homogeneity/Nonnegative homogeneity:^[2]${\displaystyle p(rx)=rp(x)}$ for all real${\displaystyle r\ge 0}$ and all${\displaystyle x\in X.}$
   • This condition holds if and only if${\displaystyle p(rx)\le rp(x)}$ for all positive real${\displaystyle r>0}$ and all${\displaystyle x\in X.}$
2. Subadditivity/Triangle inequality:^[2]${\displaystyle p(x+y)\le p(x)+p(y)}$ for all${\displaystyle x,y\in X.}$
   • This subadditivity condition requires${\displaystyle p}$ to be real-valued.

A function${\displaystyle p:X\to \mathbb{R}}$ is calledpositive^[3] ornonnegative if${\displaystyle p(x)\ge 0}$ for all${\displaystyle x\in X,}$ although some authors^[4] definepositive to instead mean that${\displaystyle p(x)\ne 0}$ whenever${\displaystyle x\ne 0;}$ these definitions are not equivalent. It is asymmetric function if${\displaystyle p(-x)=p(x)}$ for all${\displaystyle x\in X.}$ Every subadditive symmetric function is necessarily nonnegative.^{[proof 1]} A sublinear function on a real vector space issymmetric if and only if it is aseminorm. A sublinear function on a real or complex vector space is a seminorm if and only if it is abalanced function or equivalently, if and only if${\displaystyle p(ux)\le p(x)}$ for everyunit length scalar${\displaystyle u}$ (satisfying${\displaystyle |u|=1}$) and every${\displaystyle x\in X.}$

The set of all sublinear functions on${\displaystyle X,}$ denoted by${\displaystyle X^{\mathrm{\#}},}$ can bepartially ordered by declaring${\displaystyle p\le q}$ if and only if${\displaystyle p(x)\le q(x)}$ for all${\displaystyle x\in X.}$ A sublinear function is calledminimal if it is aminimal element of${\displaystyle X^{\mathrm{\#}}}$ under this order. A sublinear function is minimal if and only if it is a reallinear functional.^[1]

Everynorm,seminorm, and real linear functional is a sublinear function. Theidentity function${\displaystyle \mathbb{R}\to \mathbb{R}}$ on${\displaystyle X:=\mathbb{R}}$ is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation${\displaystyle x\mapsto -x.}$^[5] More generally, for any real${\displaystyle a\le b,}$ the mapis a sublinear function on${\displaystyle X:=\mathbb{R}}$ and moreover, every sublinear function${\displaystyle p:\mathbb{R}\to \mathbb{R}}$ is of this form; specifically, if${\displaystyle a:=-p(-1)}$ and${\displaystyle b:=p(1)}$ then${\displaystyle a\le b}$ and${\displaystyle p=S_{a,b}.}$

If${\displaystyle p}$ and${\displaystyle q}$ are sublinear functions on a real vector space${\displaystyle X}$ then so is the map${\displaystyle x\mapsto max\{p(x),q(x)\}.}$ More generally, if${\displaystyle \mathcal{P}}$ is any non-empty collection of sublinear functionals on a real vector space${\displaystyle X}$ and if for all${\displaystyle x\in X,}$${\displaystyle q(x):=sup\{p(x):p\in \mathcal{P}\},}$ then${\displaystyle q}$ is a sublinear functional on${\displaystyle X.}$^[5]

A function${\displaystyle p:X\to \mathbb{R}}$ which issubadditive,convex, and satisfies${\displaystyle p(0)\le 0}$ is also positively homogeneous (the latter condition${\displaystyle p(0)\le 0}$ is necessary as the example of${\displaystyle p(x):=\sqrt{x^2+1}}$ on${\displaystyle X:=\mathbb{R}}$ shows). If${\displaystyle p}$ is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming${\displaystyle p(0)\le 0}$, any two properties among subadditivity, convexity, and positive homogeneity implies the third.

Every sublinear function is aconvex function: For${\displaystyle 0\le t\le 1,}$

If${\displaystyle p:X\to \mathbb{R}}$ is a sublinear function on a vector space${\displaystyle X}$ then^{[proof 2][3]}$${\displaystyle p(0)=0\le p(x)+p(-x),}$$ for every${\displaystyle x\in X,}$ which implies that at least one of${\displaystyle p(x)}$ and${\displaystyle p(-x)}$ must be nonnegative; that is, for every${\displaystyle x\in X,}$^[3]$${\displaystyle 0\le max\{p(x),p(-x)\}.}$$Moreover, when${\displaystyle p:X\to \mathbb{R}}$ is a sublinear function on a real vector space then the map${\displaystyle q:X\to \mathbb{R}}$ defined by${\displaystyle q(x)\stackrel{\text{def}}{=}max\{p(x),p(-x)\}}$ is a seminorm.^[3]

Subadditivity of${\displaystyle p:X\to \mathbb{R}}$ guarantees that for all vectors${\displaystyle x,y\in X,}$^{[1][proof 3]}$${\displaystyle p(x)-p(y)\le p(x-y),}$$$${\displaystyle -p(x)\le p(-x),}$$so if${\displaystyle p}$ is alsosymmetric then thereverse triangle inequality will hold for all vectors${\displaystyle x,y\in X,}$$${\displaystyle |p(x)-p(y)|\le p(x-y).}$$

Defining${\displaystyle \ker p\stackrel{\text{def}}{=}{p}^{-1}(0),}$ then subadditivity also guarantees that for all${\displaystyle x\in X,}$ the value of${\displaystyle p}$ on the set${\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}}$ is constant and equal to${\displaystyle p(x).}$^{[proof 4]} In particular, if${\displaystyle \ker p=p^{-1}(0)}$ is a vector subspace of${\displaystyle X}$ then${\displaystyle -\ker p=\ker p}$ and the assignment${\displaystyle x+\ker p\mapsto p(x),}$ which will be denoted by${\displaystyle \hat{p},}$ is a well-defined real-valued sublinear function on thequotient space${\displaystyle X/\ker p}$ that satisfies${\displaystyle {\hat{p}}^{-1}(0)=\ker p.}$ If${\displaystyle p}$ is a seminorm then${\displaystyle \hat{p}}$ is just the usual canonical norm on the quotient space${\displaystyle X/\ker p.}$

Adding${\displaystyle bc}$ to both sides of the hypothesis (where${\displaystyle p(x+aK)\stackrel{\text{def}}{=}\{p(x+ak):k\in K\}}$) and combining that with the conclusion gives which yields many more inequalities, including, for instance, in which an expression on one side of a strict inequality can be obtained from the other by replacing the symbol${\displaystyle c}$ with${\displaystyle \mathbf{z}}$ (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

If${\displaystyle p:X\to \mathbb{R}}$ is a real-valued sublinear function on a real vector space${\displaystyle X}$ (or if${\displaystyle X}$ is complex, then when it is considered as a real vector space) then the map${\displaystyle q(x)\stackrel{\text{def}}{=}max\{p(x),p(-x)\}}$ defines aseminorm on the real vector space${\displaystyle X}$ called theseminorm associated with${\displaystyle p.}$^[3] A sublinear function${\displaystyle p}$ on a real or complex vector space is asymmetric function if and only if${\displaystyle p=q}$ where${\displaystyle q(x)\stackrel{\text{def}}{=}max\{p(x),p(-x)\}}$ as before.

More generally, if${\displaystyle p:X\to \mathbb{R}}$ is a real-valued sublinear function on a (real or complex) vector space${\displaystyle X}$ then$${\displaystyle q(x)\stackrel{\text{def}}{=}\underset{|u|=1}{sup}p(ux)=sup\{p(ux):u\text{ is a unit scalar }\}}$$ will define aseminorm on${\displaystyle X}$ if this supremum is always a real number (that is, never equal to${\displaystyle \mathrm{\infty }}$).

If${\displaystyle p}$ is a sublinear function on a real vector space${\displaystyle X}$ then the following are equivalent:^[1]

1. ${\displaystyle p}$ is alinear functional.
2. for every${\displaystyle x\in X,}$${\displaystyle p(x)+p(-x)\le 0.}$
3. for every${\displaystyle x\in X,}$${\displaystyle p(x)+p(-x)=0.}$
4. ${\displaystyle p}$ is a minimal sublinear function.

If${\displaystyle p}$ is a sublinear function on a real vector space${\displaystyle X}$ then there exists a linear functional${\displaystyle f}$ on${\displaystyle X}$ such that${\displaystyle f\le p.}$^[1]

If${\displaystyle X}$ is a real vector space,${\displaystyle f}$ is a linear functional on${\displaystyle X,}$ and${\displaystyle p}$ is a positive sublinear function on${\displaystyle X,}$ then${\displaystyle f\le p}$ on${\displaystyle X}$ if and only if^[1]

A real-valued function${\displaystyle f}$ defined on a subset of a real or complex vector space${\displaystyle X}$ is said to bedominated by a sublinear function${\displaystyle p}$ if${\displaystyle f(x)\le p(x)}$ for every${\displaystyle x}$ that belongs to the domain of${\displaystyle f.}$ If${\displaystyle f:X\to \mathbb{R}}$ is a reallinear functional on${\displaystyle X}$ then^[6][1]${\displaystyle f}$ is dominated by${\displaystyle p}$ (that is,${\displaystyle f\le p}$) if and only if$${\displaystyle -p(-x)\le f(x)\le p(x)\text{ for every }x\in X.}$$ Moreover, if${\displaystyle p}$ is a seminorm or some othersymmetric map (which by definition means that${\displaystyle p(-x)=p(x)}$ holds for all${\displaystyle x}$) then${\displaystyle f\le p}$ if and only if${\displaystyle |f|\le p.}$

Suppose${\displaystyle X}$ is atopological vector space (TVS) over the real or complex numbers and${\displaystyle p}$ is a sublinear function on${\displaystyle X.}$ Then the following are equivalent:^[7]

1. ${\displaystyle p}$ is continuous;
2. ${\displaystyle p}$ is continuous at 0;
3. ${\displaystyle p}$ is uniformly continuous on${\displaystyle X}$;

and if${\displaystyle p}$ is positive then this list may be extended to include:

4. is open in${\displaystyle X.}$

If${\displaystyle X}$ is a real TVS,${\displaystyle f}$ is a linear functional on${\displaystyle X,}$ and${\displaystyle p}$ is a continuous sublinear function on${\displaystyle X,}$ then${\displaystyle f\le p}$ on${\displaystyle X}$ implies that${\displaystyle f}$ is continuous.^[7]

The concept can be extended to operators that are homogeneous and subadditive. This requires only that thecodomain be, say, anordered vector space to make sense of the conditions.

Incomputer science, a function${\displaystyle f:{\mathbb{Z}}^{+}\to \mathbb{R}}$ is calledsublinear if${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{n}=0,}$ or${\displaystyle f(n)\in o(n)}$ inasymptotic notation (notice the small${\displaystyle o}$). Formally,${\displaystyle f(n)\in o(n)}$ if and only if, for any given${\displaystyle c>0,}$ there exists an${\displaystyle N}$ such that for${\displaystyle n\ge N.}$^[8]That is,${\displaystyle f}$ grows slower than any linear function.The two meanings should not be confused: while a Banach functional isconvex, almost the opposite is true for functions of sublinear growth: every function${\displaystyle f(n)\in o(n)}$ can be upper-bounded by aconcave function of sublinear growth.^[9]

• Asymmetric norm – Generalization of the concept of a norm
• Auxiliary normed space
• Hahn-Banach theorem – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets
• Linear functional – Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets
• Minkowski functional – Function made from a set
• Norm (mathematics) – Length in a vector space
• Seminorm – Mathematical function
• Superadditivity – Property of a function

Proofs

• Kubrusly, Carlos S. (2011).The Elements of Operator Theory (Second ed.). Boston:Birkhäuser.ISBN 978-0-8176-4998-2.OCLC 710154895.
• 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.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• Schechter, Eric (1996).Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press.ISBN 978-0-12-622760-4.OCLC 175294365.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

• Functional analysis
• Linear algebra
• Types of functions

• CS1 maint: multiple names: authors list
• CS1 maint: location missing publisher
• Articles with short description
• Short description matches Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link
• Articles containing proofs