In
mathematics, particularly 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 (e.g. inner product, norm, topology, etc.) and the linear functions defined on ...
, a seminorm is a
vector space norm that need not be
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fun ...
. Seminorms are intimately connected with
convex sets: every seminorm is the
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, the ...
of some
absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
A
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
is locally convex if and only if its topology is induced by a family of seminorms.
Definition
Let
be a vector space over 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s
or the
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers
A
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real fu ...
is called a if it satisfies the following two conditions:
#
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
/
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
:
for all
#
Absolute homogeneity:
for all
and all scalars
These two conditions imply that
[If denotes the zero vector in while denote the zero scalar, then absolute homogeneity implies that ] and that every seminorm
also has the following property:
[Suppose is a seminorm and let Then absolute homogeneity implies The triangle inequality now implies Because was an arbitrary vector in it follows that which implies that (by subtracting from both sides). Thus which implies (by multiplying thru by ).]
- Nonnegativity: for all
Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.
By definition, a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
on
is a seminorm that also separates points, meaning that it has the following additional property:
Positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fun ...
/: for all if then
A is a pair
consisting of a vector space
and a seminorm
on
If the seminorm
is also a norm then the seminormed space
is called a .
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a
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 is a real-valued function with only some of the properties of a seminorm ...
. A map
is called a if it is subadditive and
positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
.
A real-valued function
is a seminorm if and only if it is a
sublinear 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 is a real-valued function with only some of the properties of a seminorm ...
and
balanced function.
Examples
- The on which refers to the constant map on induces the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
on
- If is any
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , ...
on a vector space then its absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
defined by is a seminorm.
- A
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 is a real-valued function with only some of the properties of a seminorm ...
on a real vector space is a seminorm if and only if it is a , meaning that for all
- Every real-valued
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 is a real-valued function with only some of the properties of a seminorm ...
on a real vector space induces a seminorm defined by
- Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a
vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
is once again a seminorm (respectively, norm).
- If and are seminorms (respectively, norms) on and then the map defined by is a seminorm (respectively, a norm) on In particular, the maps on defined by and are both seminorms on
- If and are seminorms on then so are
where and
- The space of seminorms on is generally not a
distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...
with respect to the above operations. For example, over , are such that
- If is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
and is a seminorm on then is a seminorm on The seminorm will be a norm on if and only if is injective and the restriction is a norm on
Minkowski functionals and seminorms
Seminorms on a vector space
are intimately tied, via Minkowski functionals, to subsets of
that are
convex,
balanced, and
absorbing. Given such a subset
of
the Minkowski functional of
is a seminorm. Conversely, given a seminorm
on
the sets
and
are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is
Algebraic properties
Every seminorm is a
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 is a real-valued function with only some of the properties of a seminorm ...
, and thus satisfies all
properties of a sublinear function, including:
*
Convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope, ...
*
Reverse triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
:
* For any
,
* For any
,
is an
absorbing disk in
*
*
and
* 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 sublinear function on
then
on
if and only if
Other properties of seminorms
Every seminorm is a
balanced function.
If
If__is_a_set_satisfying__then__is__absorbing_in__and__where__denotes_the_Minkowski_functional_
In__mathematics,_in_the_field_of_functional_analysis,_a_Minkowski_functional_(after_Hermann_Minkowski)_or_gauge_function_is_a_function_that_recovers_a_notion_of_distance_on_a_linear_space.
If_K_is_a_subset_of_a_real_or_complex_vector_space_X,_the_...