Are All Sublinear Functions Uniformly Continuous

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 ${\displaystyle X}$ 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.^[1]

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

Definitions [edit]

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 real-valued function ${\displaystyle p:X\to \mathbb {R} }$ on ${\displaystyle X}$ is called a sublinear function (or a sublinear functional if ${\displaystyle \mathbb {K} =\mathbb {R} }$ ), and also sometimes called a quasi-seminorm or a Banach functional , if it has these two properties:^[1]

1. Positive homogeneity/Nonnegative homogeneity:^[2] ${\displaystyle p(rx)=rp(x)}$ for all real ${\displaystyle r\geq 0}$ and all ${\displaystyle x\in X.}$
2. Subadditivity/Triangle inequality:^[2] ${\displaystyle p(x+y)\leq p(x)+p(y)}$ 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 called positive ^[3] or nonnegative if ${\displaystyle p(x)\geq 0}$ for all ${\displaystyle x\in X.}$ 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 ${\displaystyle u}$ (satisfying ${\displaystyle |u|=1}$ ) and every ${\displaystyle x\in X.}$

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 [edit]

Every norm, seminorm, and real linear functional is a sublinear function. The identity 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 ${\displaystyle x\mapsto -x.}$ ^[4] More generally, for any real ${\displaystyle a\leq b,}$ the map

$${\displaystyle {\begin{alignedat}{4}S_{a,b}:\;&&\mathbb {R} &&\;\to \;&\mathbb {R} \\[0.3ex]&&x&&\;\mapsto \;&{\begin{cases}ax&{\text{ if }}x\leq 0\\bx&{\text{ if }}x\geq 0\\\end{cases}}\\\end{alignedat}}}$$

is a sublinear function on ${\displaystyle X:=\mathbb {R} }$ 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 ${\displaystyle a\leq b}$ 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 ${\displaystyle X.}$ ^[4]

A function ${\displaystyle p:X\to \mathbb {R} }$ is sublinear if and only if it is subadditive, convex, and satisfies ${\displaystyle p(0)\leq 0.}$

Properties [edit]

Every sublinear function is a convex function.

If ${\displaystyle p:X\to \mathbb {R} }$ is a sublinear function on a vector space ${\displaystyle X}$ then^{[proof 2] [3]}

$${\displaystyle p(0)~=~0~\leq ~p(x)+p(-x)\qquad {\text{ for every }}x\in X,}$$

which implies that at least one of ${\displaystyle p(x)}$ and ${\displaystyle p(-x)}$ must be nonnegative; that is,^[3]

$${\displaystyle 0~\leq ~\max\{p(x),p(-x)\}\qquad {\text{ for every }}x\in X.}$$

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 ${\displaystyle q(x):=\max\{p(x),p(-x)\}}$ is a seminorm.^[3]

Subadditivity of ${\displaystyle p:X\to \mathbb {R} }$ guarantees^{[1] [proof 3]}

$${\displaystyle p(x)-p(y)~\leq ~p(x-y)\qquad {\text{ for all }}x,y\in X}$$

so if ${\displaystyle p}$ is also symmetric then the reverse triangle inequality will hold:

$${\displaystyle |p(x)-p(y)|~\leq ~p(x-y)\qquad {\text{ for all }}x,y\in X.}$$

Pryce's Sublinearity Lemma ^[2]  —Suppose ${\displaystyle p:X\to \mathbb {R} }$ is a sublinear functional on a vector space ${\displaystyle X}$ and that ${\displaystyle C\subseteq X}$ is a non-empty convex subset. If ${\displaystyle x\in X}$ is a vector and ${\displaystyle r_{0},R>0}$ are positive real numbers such that

$${\displaystyle p(x)+r_{0}R~<~\inf _{c\in C}p\left(x+r_{0}c\right)}$$

then for every positive real ${\displaystyle t>0}$ there exists some ${\displaystyle c_{0}\in C}$ such that

$${\displaystyle p\left(x+r_{0}c_{0}\right)+tR~<~\inf _{c\in C}p\left(x+r_{0}c_{0}+tc\right).}$$

Associated seminorm [edit]

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 ${\displaystyle q(x):=\max\{p(x),p(-x)\}}$ defines a seminorm on the real vector space ${\displaystyle X}$ called the seminorm associated with ${\displaystyle p.}$ ^[3] A sublinear function ${\displaystyle p}$ on a real or complex vector space is a symmetric function if and only if ${\displaystyle p=q}$ where ${\displaystyle q(x):=\max\{p(x),p(-x)\}}$ 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

$${\displaystyle q(x)~:=~\sup _{|u|=1}p(ux)~=~\sup\{p(ux):u{\text{ is a unit scalar }}\}}$$

will define a seminorm on ${\displaystyle X}$ if this supremum is always a real number (that is, never equal to ${\displaystyle \infty }$ ).

Relation to linear functionals [edit]

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

1. ${\displaystyle p}$ is a linear functional.
2. for every ${\displaystyle x\in X,}$ ${\displaystyle p(x)+p(-x)\leq 0.}$
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 ${\displaystyle f\leq p.}$ ^[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 ${\displaystyle f\leq p}$ on ${\displaystyle X}$ if and only if Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle f^{-1}(1) \cap \{x \in X : p(x) < 1\} = \varnothing.} ^[1]

Dominating a linear functional [edit]

A real-valued function ${\displaystyle f}$ defined on a subset of a real or complex vector space ${\displaystyle X}$ is said to be dominated by a sublinear function ${\displaystyle p}$ if ${\displaystyle f(x)\leq p(x)}$ for every ${\displaystyle x}$ that belongs to the domain of ${\displaystyle f.}$ If ${\displaystyle f:X\to \mathbb {R} }$ is a real linear functional on ${\displaystyle X}$ then^{[5] [1]} ${\displaystyle f}$ is dominated by ${\displaystyle p}$ (that is, ${\displaystyle f\leq p}$ ) if and only if

$${\displaystyle -p(-x)\leq f(x)\leq p(x)\quad {\text{ for every }}x\in X.}$$

Moreover, if ${\displaystyle p}$ is a seminorm or some other symmetric map (which by definition means that ${\displaystyle p(-x)=p(x)}$ holds for all ${\displaystyle x}$ ) then ${\displaystyle f\leq p}$ if and only if ${\displaystyle |f|\leq p.}$

Theorem^[1]  —If ${\displaystyle p:X\to \mathbb {R} }$ be a sublinear function on a real vector space ${\displaystyle X}$ and if ${\displaystyle z\in X}$ then there exists a linear functional ${\displaystyle f}$ on ${\displaystyle X}$ that is dominated by ${\displaystyle p}$ (that is, ${\displaystyle f\leq p}$ ) and satisfies ${\displaystyle f(z)=p(z).}$ Moreover, if ${\displaystyle X}$ is a topological vector space and ${\displaystyle p}$ is continuous at the origin then ${\displaystyle f}$ is continuous.

Continuity [edit]

Theorem^[6]  —Suppose ${\displaystyle f:X\to \mathbb {R} }$ is a subadditive function (that is, ${\displaystyle f(x+y)\leq f(x)+f(y)}$ for all ${\displaystyle x,y\in X}$ ). Then ${\displaystyle f}$ is continuous at the origin if and only if ${\displaystyle f}$ is uniformly continuous on ${\displaystyle X.}$ If ${\displaystyle f}$ satisfies ${\displaystyle f(0)=0}$ then ${\displaystyle f}$ is continuous if and only if its absolute value ${\displaystyle |f|:X\to [0,\infty )}$ is continuous. If ${\displaystyle f}$ is non-negative then ${\displaystyle f}$ is continuous if and only if ${\displaystyle \{x\in X:f(x)<1\}}$ is open in ${\displaystyle X.}$

Suppose ${\displaystyle X}$ is a topological vector space (TVS) over the real or complex numbers and ${\displaystyle p}$ is a sublinear function on ${\displaystyle X.}$ Then the following are equivalent:^[6]

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 we may add to this list:

4. ${\displaystyle \{x\in X:p(x)<1\}}$ 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 ${\displaystyle f\leq p}$ on ${\displaystyle X}$ implies that ${\displaystyle f}$ is continuous.^[6]

Relation to Minkowski functions and open convex sets [edit]

Theorem^[6]  —If ${\displaystyle U}$ is a convex open neighborhood of the origin in a TVS ${\displaystyle X}$ then the Minkowski functional of ${\displaystyle U,}$ ${\displaystyle p_{U}:X\to [0,\infty ),}$ is a continuous non-negative sublinear function on ${\displaystyle X}$ such that ${\displaystyle U=\left\{x\in X:p_{U}(x)<1\right\};}$ if in addition ${\displaystyle U}$ is balanced then ${\displaystyle p_{U}}$ is a seminorm on ${\displaystyle X.}$

Relation to open convex sets [edit]

Theorem^[6]  —Suppose that ${\displaystyle X}$ is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of ${\displaystyle X}$ are exactly those that are of the form

$${\displaystyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}}$$

for some ${\displaystyle z\in X}$ and some positive continuous sublinear function ${\displaystyle p}$ on ${\displaystyle X.}$

Operators [edit]

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 [edit]

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 ${\displaystyle n\geq N.}$ ^[7] 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.^[8]

See also [edit]

• Asymmetric norm – Generalization of the concept of a norm
• Auxiliary normed space
• Hahn-Banach theorem
• Linear functional
• Minkowski functional
• Norm (mathematics) – Length in a vector space
• Seminorm
• Superadditivity

Notes [edit]

References [edit]

1. ^ ^{a b c d e f g h i} Narici & Beckenstein 2011, pp. 177–220.
2. ^ ^{a b c} Schechter 1996, pp. 313–315.
3. ^ ^{a b c d e} Narici & Beckenstein 2011, pp. 120–121.
4. ^ ^{a b} Narici & Beckenstein 2011, pp. 177–221.
5. ^ Rudin 1991, pp. 56–62.
6. ^ ^{a b c d e} Narici & Beckenstein 2011, pp. 192–193.
7. ^ Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001) [1990]. "3.1". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 47–48. ISBN0-262-03293-7. {{cite book}}: CS1 maint: multiple names: authors list (link)
8. ^ Ceccherini-Silberstein, Tullio; Salvatori, Maura; Sava-Huss, Ecaterina (2017-06-29). Groups, graphs, and random walks. Cambridge. Lemma 5.17. ISBN9781316604403. OCLC 948670194.

Bibliography [edit]

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN978-0-07-054236-5. OCLC 21163277.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. 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. ISBN978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN978-0-12-622760-4. OCLC 175294365.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN978-0-486-45352-1. OCLC 853623322.

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Source: https://en.wikipedia.org/wiki/Sublinear_function

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