In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
, a sublinear function (or
functional as is more often used in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
), also called a quasi-seminorm or a Banach functional, on a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
is a
real-valued
function with only some of the properties of a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
. 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 that it is not required to map non-zero vectors to non-zero values.
In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, described below, that also goes by the name "sublinear function."
Definitions
Let
be a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a field
where
is either the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
A real-valued function
on
is called a ' (or a ' if
), and also sometimes called a ' or a ', if it has these two properties:
- '' Positive homogeneity/ Nonnegative homogeneity'': for all real and all
* This condition holds if and only if for all positive real and all
- '' Subadditivity/ Triangle inequality'': for all
* This subadditivity condition requires to be real-valued.
A function
is called or if
for all
although some authors define to instead mean that
whenever
these definitions are not equivalent.
It is a if
for all
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
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
.
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 ' 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.
Examples and sufficient conditions
Every norm, seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
, 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
More generally, for any real the map
is a sublinear function on and moreover, every sublinear function is of this form; specifically, if and then and
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
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.
Properties
Every sublinear function is a convex function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
: For
If is a sublinear function on a vector space then[
for every which implies that at least one of and must be nonnegative; that is, for every
Moreover, when is a sublinear function on a real vector space then the map defined by is a seminorm.
Subadditivity of guarantees that for all vectors ][
so if is also symmetric then the reverse triangle inequality will hold for all vectors
Defining then subadditivity also guarantees that for all the value of on the set is constant and equal to ][
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
Adding to both sides of the hypothesis (where ) 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 with (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).
]
Associated seminorm
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
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on the real vector space called the seminorm associated with
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
will define a seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on if this supremum is always a real number (that is, never equal to ).
Relation to linear functionals
If is a sublinear function on a real vector space then the following are equivalent:
- 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
If is a real vector space, is a linear functional on and is a positive sublinear function on then on if and only if
Dominating a linear functional
A real-valued function defined on a subset of a real or complex vector space is said to be a sublinear function if for every that belongs to the domain of
If is a real linear functional on then is dominated by (that is, ) if and only if
Moreover, if is a seminorm or some other (which by definition means that holds for all ) then if and only if
Continuity
Suppose is a topological vector space (TVS) over the real or complex numbers and is a sublinear function on
Then the following are equivalent:
- 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.
Relation to Minkowski functions and open convex sets
Relation to open convex sets
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
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied 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
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.
See also
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Notes
Proofs
References
Bibliography
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{{Topological vector spaces
Articles containing proofs
Functional analysis
Linear algebra
Types of functions