Sublinear function

In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X {\displaystyle X} is a real-valued function with only 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 his version of the Hahn-Banach theorem.
There is also a different notion in computer science, described below, that also goes by the name "sublinear function."
Let X {\displaystyle X} be a vector space over a field K , {\displaystyle \mathbb {K} ,} where K {\displaystyle \mathbb {K} } is either the real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } on X {\displaystyle X} is called a sublinear function (or a sublinear functional if K = R {\displaystyle \mathbb {K} =\mathbb {R} } ), and also sometimes called a quasi-seminorm or a Banach functional, if it has these two properties:
A function p : X → R {\displaystyle p:X\to \mathbb {R} } is called positive or nonnegative if p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X , {\displaystyle x\in X,} although some authors define positive to instead mean that p ( x ) ≠ 0 {\displaystyle p(x)\neq 0} whenever x ≠ 0 ; {\displaystyle x\neq 0;} these definitions are not equivalent. It is a symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X . {\displaystyle x\in X.} Every subadditive symmetric function is necessarily nonnegative. 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 p ( u x ) ≤ p ( x ) {\displaystyle p(ux)\leq p(x)} for every unit length scalar u {\displaystyle u} (satisfying | u | = 1 {\displaystyle |u|=1} ) and every x ∈ X . {\displaystyle x\in X.}
The set of all sublinear functions on X , {\displaystyle X,} denoted by X # , {\displaystyle X^{\#},} can be partially ordered by declaring p ≤ q {\displaystyle p\leq q} if and only if p ( x ) ≤ q ( x ) {\displaystyle p(x)\leq q(x)} for all x ∈ X . {\displaystyle x\in X.} A sublinear function is called minimal if it is a minimal element of X # {\displaystyle X^{\#}} under this order. A sublinear function is minimal if and only if it is a real linear functional.
Every norm, seminorm, and real linear functional is a sublinear function. The identity function R → R {\displaystyle \mathbb {R} \to \mathbb {R} } on X := R {\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 x ↦ − x . {\displaystyle x\mapsto -x.} More generally, for any real a ≤ b , {\displaystyle a\leq b,} the map S a , b : R → R x ↦ { a x if x ≤ 0 b x if x ≥ 0 {\displaystyle {\begin{alignedat}{4}S_{a,b}:\;&&\mathbb {R} &&\;\to \;&\mathbb {R} \\[0.3ex]&&x&&\;\mapsto \;&{\begin{cases}ax&{\text{ if }}x\leq 0\\bx&{\text{ if }}x\geq 0\\\end{cases}}\\\end{alignedat}}} is a sublinear function on X := R {\displaystyle X:=\mathbb {R} } and moreover, every sublinear function p : R → R {\displaystyle p:\mathbb {R} \to \mathbb {R} } is of this form; specifically, if a := − p ( − 1 ) {\displaystyle a:=-p(-1)} and b := p ( 1 ) {\displaystyle b:=p(1)} then a ≤ b {\displaystyle a\leq b} and p = S a , b . {\displaystyle p=S_{a,b}.}
If p {\displaystyle p} and q {\displaystyle q} are sublinear functions on a real vector space X {\displaystyle X} then so is the map x ↦ max { p ( x ) , q ( x ) } . {\displaystyle x\mapsto \max\{p(x),q(x)\}.} More generally, if P {\displaystyle {\mathcal {P}}} is any non-empty collection of sublinear functionals on a real vector space X {\displaystyle X} and if for all x ∈ X , {\displaystyle x\in X,} q ( x ) := sup { p ( x ) : p ∈ P } , {\displaystyle q(x):=\sup\{p(x):p\in {\mathcal {P}}\},} then q {\displaystyle q} is a sublinear functional on X . {\displaystyle X.}
A function p : X → R {\displaystyle p:X\to \mathbb {R} } which is subadditive, convex, and satisfies p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is also positively homogeneous (the latter condition p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} is necessary as the example of p ( x ) := x 2 + 1 {\displaystyle p(x):={\sqrt {x^{2}+1}}} on X := R {\displaystyle X:=\mathbb {R} } shows). If p {\displaystyle p} is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} , any two properties among subadditivity, convexity, and positive homogeneity implies the third.
Every sublinear function is a convex function: For 0 ≤ t ≤ 1 , {\displaystyle 0\leq t\leq 1,} p ( t x + ( 1 − t ) y ) ≤ p ( t x ) + p ( ( 1 − t ) y ) subadditivity = t p ( x ) + ( 1 − t ) p ( y ) nonnegative homogeneity {\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 p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function on a vector space X {\displaystyle X} then p ( 0 ) = 0 ≤ p ( x ) + p ( − x ) , {\displaystyle p(0)~=~0~\leq ~p(x)+p(-x),} for every x ∈ X , {\displaystyle x\in X,} which implies that at least one of p ( x ) {\displaystyle p(x)} and p ( − x ) {\displaystyle p(-x)} must be nonnegative; that is, for every x ∈ X , {\displaystyle x\in X,} 0 ≤ max { p ( x ) , p ( − x ) } . {\displaystyle 0~\leq ~\max\{p(x),p(-x)\}.} Moreover, when p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function on a real vector space then the map q : X → R {\displaystyle q:X\to \mathbb {R} } defined by q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} is a seminorm.
Subadditivity of p : X → R {\displaystyle p:X\to \mathbb {R} } guarantees that for all vectors x , y ∈ X , {\displaystyle x,y\in X,} p ( x ) − p ( y ) ≤ p ( x − y ) , {\displaystyle p(x)-p(y)~\leq ~p(x-y),} − p ( x ) ≤ p ( − x ) , {\displaystyle -p(x)~\leq ~p(-x),} so if p {\displaystyle p} is also symmetric then the reverse triangle inequality will hold for all vectors x , y ∈ X , {\displaystyle x,y\in X,} | p ( x ) − p ( y ) | ≤ p ( x − y ) . {\displaystyle |p(x)-p(y)|~\leq ~p(x-y).}
Defining ker ⁡ p = def p − 1 ( 0 ) , {\displaystyle \ker p~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~p^{-1}(0),} then subadditivity also guarantees that for all x ∈ X , {\displaystyle x\in X,} the value of p {\displaystyle p} on the set x + ( ker ⁡ p ∩ − ker ⁡ p ) = { x + k : p ( k ) = 0 = p ( − k ) } {\displaystyle x+(\ker p\cap -\ker p)=\{x+k:p(k)=0=p(-k)\}} is constant and equal to p ( x ) . {\displaystyle p(x).} In particular, if ker ⁡ p = p − 1 ( 0 ) {\displaystyle \ker p=p^{-1}(0)} is a vector subspace of X {\displaystyle X} then − ker ⁡ p = ker ⁡ p {\displaystyle -\ker p=\ker p} and the assignment x + ker ⁡ p ↦ p ( x ) , {\displaystyle x+\ker p\mapsto p(x),} which will be denoted by p ^ , {\displaystyle {\hat {p}},} is a well-defined real-valued sublinear function on the quotient space X / ker ⁡ p {\displaystyle X\,/\,\ker p} that satisfies p ^ − 1 ( 0 ) = ker ⁡ p . {\displaystyle {\hat {p}}^{-1}(0)=\ker p.} If p {\displaystyle p} is a seminorm then p ^ {\displaystyle {\hat {p}}} is just the usual canonical norm on the quotient space X / ker ⁡ p . {\displaystyle X\,/\,\ker p.}
Adding b c {\displaystyle bc} to both sides of the hypothesis p ( x ) + a c < inf p ( x + a K ) {\textstyle p(x)+ac\,<\,\inf _{}p(x+aK)} (where p ( x + a K ) = def { p ( x + a k ) : k ∈ K } {\displaystyle p(x+aK)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{p(x+ak):k\in K\}} ) and combining that with the conclusion gives p ( x ) + a c + b c < inf p ( x + a K ) + b c ≤ p ( x + a z ) + b c < inf p ( x + a z + b K ) {\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, p ( x ) + a c + b c < p ( x + a z ) + b c < p ( x + a z + b z ) {\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 c {\displaystyle c} with z {\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 p : X → R {\displaystyle p:X\to \mathbb {R} } is a real-valued sublinear function on a real vector space X {\displaystyle X} (or if X {\displaystyle X} is complex, then when it is considered as a real vector space) then the map q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} defines a seminorm on the real vector space X {\displaystyle X} called the seminorm associated with p . {\displaystyle p.} A sublinear function p {\displaystyle p} on a real or complex vector space is a symmetric function if and only if p = q {\displaystyle p=q} where q ( x ) = def max { p ( x ) , p ( − x ) } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\max\{p(x),p(-x)\}} as before.
More generally, if p : X → R {\displaystyle p:X\to \mathbb {R} } is a real-valued sublinear function on a (real or complex) vector space X {\displaystyle X} then q ( x ) = def sup | u | = 1 p ( u x ) = sup { p ( u x ) : u is a unit scalar } {\displaystyle q(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup _{|u|=1}p(ux)~=~\sup\{p(ux):u{\text{ is a unit scalar }}\}} will define a seminorm on X {\displaystyle X} if this supremum is always a real number (that is, never equal to ∞ {\displaystyle \infty } ).
If p {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} then the following are equivalent:
If p {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} then there exists a linear functional f {\displaystyle f} on X {\displaystyle X} such that f ≤ p . {\displaystyle f\leq p.}
If X {\displaystyle X} is a real vector space, f {\displaystyle f} is a linear functional on X , {\displaystyle X,} and p {\displaystyle p} is a positive sublinear function on X , {\displaystyle X,} then f ≤ p {\displaystyle f\leq p} on X {\displaystyle X} if and only if f − 1 ( 1 ) ∩ { x ∈ X : p ( x ) < 1 } = ∅ . {\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .}
A real-valued function f {\displaystyle f} defined on a subset of a real or complex vector space X {\displaystyle X} is said to be dominated by a sublinear function p {\displaystyle p} if f ( x ) ≤ p ( x ) {\displaystyle f(x)\leq p(x)} for every x {\displaystyle x} that belongs to the domain of f . {\displaystyle f.} If f : X → R {\displaystyle f:X\to \mathbb {R} } is a real linear functional on X {\displaystyle X} then f {\displaystyle f} is dominated by p {\displaystyle p} (that is, f ≤ p {\displaystyle f\leq p} ) if and only if − p ( − x ) ≤ f ( x ) ≤ p ( x ) for every x ∈ X . {\displaystyle -p(-x)\leq f(x)\leq p(x)\quad {\text{ for every }}x\in X.} Moreover, if p {\displaystyle p} is a seminorm or some other symmetric map (which by definition means that p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} holds for all x {\displaystyle x} ) then f ≤ p {\displaystyle f\leq p} if and only if | f | ≤ p . {\displaystyle |f|\leq p.}
Suppose X {\displaystyle X} is a topological vector space (TVS) over the real or complex numbers and p {\displaystyle p} is a sublinear function on X . {\displaystyle X.} Then the following are equivalent:
and if p {\displaystyle p} is positive then this list may be extended to include:
If X {\displaystyle X} is a real TVS, f {\displaystyle f} is a linear functional on X , {\displaystyle X,} and p {\displaystyle p} is a continuous sublinear function on X , {\displaystyle X,} then f ≤ p {\displaystyle f\leq p} on X {\displaystyle X} implies that f {\displaystyle f} is continuous.
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.
In computer science, a function f : Z + → R {\displaystyle f:\mathbb {Z} ^{+}\to \mathbb {R} } is called sublinear if lim n → ∞ f ( n ) n = 0 , {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{n}}=0,} or f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} in asymptotic notation (notice the small o {\displaystyle o} ). Formally, f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} if and only if, for any given c > 0 , {\displaystyle c>0,} there exists an N {\displaystyle N} such that f ( n ) < c n {\displaystyle f(n)<cn} for n ≥ N . {\displaystyle n\geq N.} That is, f {\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 f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} can be upper-bounded by a concave function of sublinear growth.
Proofs