Sublinear function

In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm except that it is not required to map non-zero vectors to non-zero values.

In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved the Hahn-Banach theorem.^[1]

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

Definitions

Let ${\displaystyle X}$ be a vector space over a field ${\displaystyle \mathbb {K} ,}$ where ${\displaystyle \mathbb {K} }$ is either the real numbers ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} .}$ A function ${\displaystyle p\colon X\to \mathbb {R} }$ is called a sublinear if it has these two properties:^[1]

1. Positive homogeneity,^[2] that is ${\displaystyle p(rx)=rp(x)}$, for all ${\displaystyle r\geq 0}$ and ${\displaystyle x\in X}$.
2. Subadditivity,^[2] that is ${\displaystyle p(x+y)\leq p(x)+p(y)}$ for ${\displaystyle x,y\in X.}$

A function ${\displaystyle p:X\to \mathbb {R} }$ is called positive^[3] or nonnegative if ${\displaystyle p(x)\geq 0}$ for all ${\displaystyle x\in X,}$ although some authors^[4] define positive to instead mean that ${\displaystyle p(x)\neq 0}$ whenever ${\displaystyle x\neq 0;}$ these definitions are not equivalent. It is a symmetric 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 is symmetric if and only if it is a seminorm. A sublinear function on a real or complex vector space is a seminorm if and only if it is a balanced function or equivalently, if and only if ${\displaystyle p(ux)\leq p(x)}$ for every unit length scalar ${\displaystyle u}$ and ${\displaystyle x\in X.}$

The set of all sublinear functions on ${\displaystyle X,}$ denoted by ${\displaystyle X^{\#},}$ can be partially ordered by declaring ${\displaystyle p\leq q}$ if and only if ${\displaystyle p(x)\leq q(x)}$ for all ${\displaystyle x\in X.}$ A sublinear function is called minimal if it is a minimal element of ${\displaystyle X^{\#}}$ under this order. A sublinear function is minimal if and only if it is a real linear functional.^[1]

Examples and sufficient conditions

Every norm, seminorm, and real linear functional is a sublinear function. The identity function on ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle a\leq b,}$ the map

${\displaystyle S_{a,b}\colon \mathbb {R} \to \mathbb {R} ,\quad x\mapsto {\begin{cases}ax,&{\text{if }}x\leq 0,\\bx,&{\text{if }}x\geq 0\end{cases}}}$

is a sublinear function on ${\displaystyle \mathbb {R} }$ and moreover, every sublinear function ${\displaystyle p\colon \mathbb {R} \to \mathbb {R} }$ is of this form; specifically, if ${\displaystyle a=-p(-1)}$ and ${\displaystyle b=p(1)}$ then ${\displaystyle a\leq b}$ 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 ${\displaystyle p\colon X\to \mathbb {R} }$ which is subadditive, convex, and satisfies ${\displaystyle p(0)\leq 0}$ is also positively homogeneous (the latter condition ${\displaystyle p(0)\leq 0}$ 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 ${\displaystyle p(0)\leq 0}$, any two properties among subadditivity, convexity, and positive homogeneity implies the third.

Properties

Every sublinear function is a convex function: For ${\displaystyle 0\leq t\leq 1,}$ $${\displaystyle {\begin{alignedat}{3}p(tx+(1-t)y)&\leq p(tx)+p((1-t)y)&&\quad {\text{ subadditivity}}\\&=tp(x)+(1-t)p(y)&&\quad {\text{ nonnegative homogeneity}}\\\end{alignedat}}}$$

If ${\displaystyle p:X\to \mathbb {R} }$ is a sublinear function on a vector space ${\displaystyle X}$ then^{[proof 2][3]} $${\displaystyle p(0)~=~0~\leq ~p(x)+p(-x),}$$ 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] $${\displaystyle 0~\leq ~\max\{p(x),p(-x)\}.}$$ 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 ${\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}}$ 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]} $${\displaystyle p(x)-p(y)~\leq ~p(x-y),}$$ $${\displaystyle -p(x)~\leq ~p(-x),}$$ so if ${\displaystyle p}$ is also symmetric then the reverse triangle inequality will hold for all vectors ${\displaystyle x,y\in X,}$ $${\displaystyle |p(x)-p(y)|~\leq ~p(x-y).}$$

Defining ${\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),}$ 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 the quotient 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 ${\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)}$ (where ${\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}}$) and combining that with the conclusion gives $${\displaystyle p(x)+ac+bc~<~\inf _{}p(x+aK)+bc~\leq ~p(x+a\mathbf {z} )+bc~<~\inf _{}p(x+a\mathbf {z} +bK)}$$ which yields many more inequalities, including, for instance, $${\displaystyle p(x)+ac+bc~<~p(x+a\mathbf {z} )+bc~<~p(x+a\mathbf {z} +b\mathbf {z} )}$$ in which an expression on one side of a strict inequality ${\displaystyle \,<\,}$ 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).

Operators

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

Computer science definition

In computer science, a function ${\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} }$ is called sublinear if ${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,}$ or ${\displaystyle f(n)\in o(n)}$ in asymptotic 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 ${\displaystyle f(n)<cn}$ for ${\displaystyle n\geq N.}$^[8] That is, ${\displaystyle f}$ grows slower than any linear function. The two meanings should not be confused: while a Banach functional is convex, almost the opposite is true for functions of sublinear growth: every function ${\displaystyle f(n)\in o(n)}$ can be upper-bounded by a concave function of sublinear growth.^[9]

See also

• 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

Notes

Proofs

References

Bibliography

• 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.