Seminorm


In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
A topological vector space 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 numbers or the complex numbers
A real-valued function is called a if it satisfies the following two conditions:
  1. Subadditivity/Triangle inequality: for all
  2. Absolute homogeneity: for all and all scalars
These two conditions imply that and that every seminorm also has the following property:

  1. Nonnegativity: for all
Some authors include non-negativity as part of the definition of "seminorm", although this is not necessary since it follows from the other two properties.
By definition, a norm on is a seminorm that also separates points, meaning that it has the following additional property:

  1. Positive definite/Positive/: whenever satisfies 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. 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.
A real-valued function is a seminorm if and only if it is a sublinear and balanced function.

Examples


  • The on which refers to the constant map on induces the indiscrete topology on
  • Let be a measure on a space. For an arbitrary constant, let be the set of all functions for which
    exists and is finite. It can be shown that is a vector space, and the functional is a seminorm on. However, it is not always a norm because does not always imply. To make a norm, quotient by the closed subspace of functions with. The resulting space,, has a norm induced by.
  • If is any linear form on a vector space then its absolute value defined by is a seminorm.
  • A sublinear function on a real vector space is a seminorm if and only if it is a, meaning that for all
  • Every real-valued sublinear function on a real vector space induces a seminorm defined by
  • Any finite sum of seminorms is a seminorm. The restriction of a seminorm to a vector subspace is once again a seminorm.
  • If and are seminorms on and then the map defined by is a seminorm on In particular, the maps on defined by and are both seminorms on
  • If and are seminorms on then so are
    and
    where and
  • The space of seminorms on is generally not a distributive lattice with respect to the above operations. For example, over, are such that
    while
  • If is a linear map 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 is

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity, and for all vectors :
the reverse triangle inequality:
and also
and
For any vector and positive real
and furthermore, is an absorbing disk in
If is a sublinear function on a real vector space then there exists a linear functional on such that and furthermore, for any linear functional on on if and only if
Other properties of seminorms
Every seminorm is a balanced function.
A seminorm is a norm on if and only if does not contain a non-trivial vector subspace.
If is a seminorm on then is a vector subspace of and for every is constant on the set and equal to
Furthermore, for any real
If is a set satisfying then is absorbing in and where denotes the Minkowski functional associated with . In particular, if is as above and is any seminorm on then if and only if
If is a normed space and then for all in the interval
Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.

Relationship to other norm-like concepts

Let be a non-negative function. The following are equivalent:

  1. is a seminorm.
  2. is a convex -seminorm.
  3. is a convex balanced G-seminorm.
If any of the above conditions hold, then the following are equivalent:

  1. is a norm;
  2. does not contain a non-trivial vector subspace.
  3. There exists a norm on with respect to which, is bounded.
If is a sublinear function on a real vector space then the following are equivalent:

  1. is a linear functional;
  2. ;
  3. ;

Inequalities involving seminorms

If are seminorms on then:

  • if and only if implies
  • If and are such that implies then for all
  • Suppose and are positive real numbers and are seminorms on such that for every if then Then
  • If is a vector space over the reals and is a non-zero linear functional on then if and only if
If is a seminorm on and is a linear functional on then:

  • on if and only if on .
  • on if and only if
  • If and are such that implies then for all

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:
A similar extension property also holds for seminorms:

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm on induces a topology, called the, via the canonical translation-invariant pseudometric ;
This topology is Hausdorff if and only if is a metric, which occurs if and only if is a norm.
This topology makes into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls centered at the origin:
as ranges over the positive reals.
Every seminormed space should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called.
Equivalently, every vector space with seminorm induces a vector space quotient where is the subspace of consisting of all vectors with Then carries a norm defined by The resulting topology, pulled back to is precisely the topology induced by
Any seminorm-induced topology makes locally convex, as follows. If is a seminorm on and call the set the ; likewise the closed ball of radius is The set of all open -balls at the origin forms a neighborhood basis of convex balanced sets that are open in the -topology on

Stronger, weaker, and equivalent seminorms

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If and are seminorms on then we say that is than and that is than if any of the following equivalent conditions holds:
  1. The topology on induced by is finer than the topology induced by
  2. If is a sequence in then in implies in
  3. If is a net in then in implies in
  4. is bounded on
  5. If then for all
  6. There exists a real such that on
The seminorms and are called if they are both weaker than each other. This happens if they satisfy any of the following conditions:

  1. The topology on induced by is the same as the topology induced by
  2. is stronger than and is stronger than
  3. If is a sequence in then if and only if
  4. There exist positive real numbers and such that

Normability and seminormability

A topological vector space is said to be a if its topology is induced by a single seminorm.
A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1.
A is a topological vector space that possesses a bounded neighborhood of the origin.
Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.
A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.
Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.
A TVS is normable if and only if it is a T1 space and admits a bounded convex neighborhood of the origin.
If is a Hausdorff locally convex TVS then the following are equivalent:

  1. is normable.
  2. is seminormable.
  3. has a bounded neighborhood of the origin.
  4. The strong dual of is normable.
  5. The strong dual of is metrizable.
Furthermore, is finite dimensional if and only if is normable.
The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial.

Topological properties


  • If is a TVS and is a continuous seminorm on then the closure of in is equal to
  • The closure of in a locally convex space whose topology is defined by a family of continuous seminorms is equal to
  • A subset in a seminormed space is bounded if and only if is bounded.
  • If is a seminormed space then the locally convex topology that induces on makes into a pseudometrizable TVS with a canonical pseudometric given by for all
  • The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial.