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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 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."

Let be a vector space over a field where is either the real numbers or complex numbers
A real-valued function on is called a * sublinear function* (or a

*Positive homogeneity*:**/**Nonnegative homogeneity^{[2]}for all real and all- This condition holds if and only if for all positive real and all

*Subadditivity*:**/**Triangle inequality^{[2]}for all- This subadditivity condition requires to be real-valued.

A function is called *positive*^{[3]} or *nonnegative* if for all although some authors^{[4]} define *positive* to instead mean that whenever these definitions are not equivalent.
It is a *symmetric function* if for all
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 for every unit length scalar (satisfying ) and every

The set of all sublinear functions on denoted by can be partially ordered by declaring if and only if for all
A sublinear function is called * minimal* if it is a minimal element of 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 on 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 ^{[5]}
More generally, for any real the map

If and are sublinear functions on a real vector space then so is the map More generally, if is any non-empty collection of sublinear functionals on a real vector space and if for all then is a sublinear functional on ^{[5]}

A function which is subadditive, convex, and satisfies is also positively homogeneous (the latter condition is necessary as the example of on shows). If is positively homogeneous, it is convex if and only if it is subadditive. Therefore, assuming , any two properties among subadditivity, convexity, and positive homogeneity implies the third.

Every sublinear function is a convex function: For

If is a sublinear function on a vector space then^{[proof 2]}^{[3]}

Subadditivity of guarantees that for all vectors ^{[1]}^{[proof 3]}

Defining then subadditivity also guarantees that for all the value of on the set is constant and equal to ^{[proof 4]}
In particular, if is a vector subspace of then and the assignment which will be denoted by is a well-defined real-valued sublinear function on the quotient space that satisfies If is a seminorm then is just the usual canonical norm on the quotient space

**Pryce's sublinearity lemma ^{[2]}** — Suppose is a sublinear functional on a vector space and that is a non-empty convex subset.
If is a vector and are positive real numbers such that

Adding to both sides of the hypothesis (where ) and combining that with the conclusion gives

If is a real-valued sublinear function on a real vector space (or if is complex, then when it is considered as a real vector space) then the map defines a seminorm on the real vector space called the **seminorm associated with **^{[3]}
A sublinear function on a real or complex vector space is a symmetric function if and only if where as before.

More generally, if is a real-valued sublinear function on a (real or complex) vector space then

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

- is a linear functional.
- for every
- for every
- is a minimal sublinear function.

If is a sublinear function on a real vector space then there exists a linear functional on such that ^{[1]}

If is a real vector space, is a linear functional on and is a positive sublinear function on then on if and only if ^{[1]}

A real-valued function defined on a subset of a real or complex vector space is said to be *dominated by* a sublinear function if for every that belongs to the domain of
If is a real linear functional on then^{[6]}^{[1]} is dominated by (that is, ) if and only if

**Theorem ^{[1]}** — If be a sublinear function on a real vector space and if then there exists a linear functional on that is dominated by (that is, ) and satisfies
Moreover, if is a topological vector space and is continuous at the origin then is continuous.

**Theorem ^{[7]}** — Suppose is a subadditive function (that is, for all ).
Then is continuous at the origin if and only if is uniformly continuous on
If satisfies then is continuous if and only if its absolute value is continuous.
If is non-negative then is continuous if and only if is open in

Suppose is a topological vector space (TVS) over the real or complex numbers and is a sublinear function on
Then the following are equivalent:^{[7]}

- is continuous;
- is continuous at 0;
- is uniformly continuous on ;

and if is positive then this list may be extended to include:

- is open in

If is a real TVS, is a linear functional on and is a continuous sublinear function on then on implies that is continuous.^{[7]}

**Theorem ^{[7]}** — If is a convex open neighborhood of the origin in a topological vector space then the Minkowski functional of is a continuous non-negative sublinear function on such that if in addition is a balanced set then is a seminorm on

**Theorem ^{[7]}** — Suppose that is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers.
Then the open convex subsets of are exactly those that are of the form

Let be an open convex subset of If then let and otherwise let be arbitrary. Let be the Minkowski functional of which is a continuous sublinear function on since is convex, absorbing, and open ( however is not necessarily a seminorm since was not assumed to be balanced). From it follows that

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 is called **sublinear** if or in asymptotic notation (notice the small ).
Formally, if and only if, for any given there exists an such that for ^{[8]}
That is, 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 can be upper-bounded by a concave 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 functionals
- Linear functional – Linear map from a vector space to its field of scalars
- Minkowski functional – Function made from a set
- Norm (mathematics) – Length in a vector space
- Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous
- Superadditivity

**Proofs**

**^**Let The triangle inequality and symmetry imply Substituting for and then subtracting from both sides proves that Thus which implies**^**If and then nonnegative homogeneity implies that Consequently, which is only possible if**^**which happens if and only if Substituting and gives which implies (positive homogeneity is not needed; the triangle inequality suffices).**^**Let and It remains to show that The triangle inequality implies Since as desired.

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Narici & Beckenstein 2011, pp. 177–220. - ^
^{a}^{b}^{c}Schechter 1996, pp. 313–315. - ^
^{a}^{b}^{c}^{d}^{e}Narici & Beckenstein 2011, pp. 120–121. **^**Kubrusly 2011, p. 200.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 177–221. **^**Rudin 1991, pp. 56–62.- ^
^{a}^{b}^{c}^{d}^{e}Narici & Beckenstein 2011, pp. 192–193. **^**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. ISBN 0-262-03293-7.`{{cite book}}`

: CS1 maint: multiple names: authors list (link)**^**Ceccherini-Silberstein, Tullio; Salvatori, Maura; Sava-Huss, Ecaterina (2017-06-29).*Groups, graphs, and random walks*. Cambridge. Lemma 5.17. ISBN 9781316604403. OCLC 948670194.`{{cite book}}`

: CS1 maint: location missing publisher (link)

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