In mathematics, particularly in functional analysis, a seminorm is a norm that 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 seminorm if it satisfies the following two conditions:

  1. Subadditivity[1]/Triangle inequality: for all
  2. Absolute homogeneity:[1] for all and all scalars

These two conditions imply that [proof 1] and that every seminorm also has the following property:[proof 2]

  1. Nonnegativity:[1] 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 on is a seminorm that also separates points, meaning that it has the following additional property:

  1. Positive definite/Positive[1]/Point-separating: whenever satisfies then

A seminormed space 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 normed space.

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map is called a sublinear function if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is not 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

Minkowski functionals and seminorms

Main article: Minkowski functional

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 [5]

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: [2][6] and also and [2][6]

For any vector and positive real [7] and furthermore, is an absorbing disk in [3]

If is a sublinear function on a real vector space then there exists a linear functional on such that [6] and furthermore, for any linear functional on on if and only if [6]

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 [proof 3]

Furthermore, for any real [3]

If is a set satisfying then is absorbing in and where denotes the Minkowski functional associated with (that is, the gauge of ).[5] In particular, if is as above and is any seminorm on then if and only if [5]

If is a normed space and then for all in the interval [8]

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.[9]

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.[10]
  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:[6]

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

Inequalities involving seminorms

If are seminorms on then:

If is a seminorm on and is a linear functional on then:

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:

If is a vector subspace of a seminormed space and if is a continuous linear functional on then may be extended to a continuous linear functional on that has the same norm as [15]

A similar extension property also holds for seminorms:

Theorem[16][12] (Extending seminorms) — If is a vector subspace of is a seminorm on and is a seminorm on such that then there exists a seminorm on such that and

Proof: Let be the convex hull of Then is an absorbing disk in and so the Minkowski functional of is a seminorm on This seminorm satisfies on and on

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm on induces a topology, called the seminorm-induced topology, 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.[4] 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 (or the closed 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 seminormable.

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 open ball of radius about the origin; likewise the closed ball of radius is The set of all open (resp. closed) -balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) 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 stronger than and that is weaker 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 [4]
  3. If is a net in then in implies in
  4. is bounded on [4]
  5. If then for all [4]
  6. There exists a real such that on [4]

The seminorms and are called equivalent if they are both weaker (or both stronger) 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 [4]
  3. If is a sequence in then if and only if
  4. There exist positive real numbers and such that

Normability and seminormability

See also: Normed space and Local boundedness § locally bounded topological vector space

A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space). A locally bounded topological vector space 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.[17] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[18] 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.[19]
  5. The strong dual of is metrizable.[19]

Furthermore, is finite dimensional if and only if is normable (here denotes endowed with the weak-* topology).

The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).[18]

Topological properties

Continuity of seminorms

If is a seminorm on a topological vector space then the following are equivalent:[5]

  1. is continuous.
  2. is continuous at 0;[3]
  3. is open in ;[3]
  4. is closed neighborhood of 0 in ;[3]
  5. is uniformly continuous on ;[3]
  6. There exists a continuous seminorm on such that [3]

In particular, if is a seminormed space then a seminorm on is continuous if and only if is dominated by a positive scalar multiple of [3]

If is a real TVS, is a linear functional on and is a continuous seminorm (or more generally, a sublinear function) on then on implies that is continuous.[6]

Continuity of linear maps

If is a map between seminormed spaces then let[15]

If is a linear map between seminormed spaces then the following are equivalent:

  1. is continuous;
  2. ;[15]
  3. There exists a real such that ;[15]
    • In this case,

If is continuous then for all [15]

The space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm This seminorm is a norm if is a norm.[15]

Generalizations

The concept of norm in composition algebras does not share the usual properties of a norm.

A composition algebra consists of an algebra over a field an involution and a quadratic form which is called the "norm". In several cases is an isotropic quadratic form so that has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An ultraseminorm or a non-Archimedean seminorm is a seminorm that also satisfies

Weakening subadditivity: Quasi-seminorms

A map is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some such that The smallest value of for which this holds is called the multiplier of

A quasi-seminorm that separates points is called a quasi-norm on

Weakening homogeneity - -seminorms

A map is called a -seminorm if it is subadditive and there exists a such that and for all and scalars A -seminorm that separates points is called a -norm on

We have the following relationship between quasi-seminorms and -seminorms:

Suppose that is a quasi-seminorm on a vector space with multiplier If then there exists -seminorm on equivalent to

See also

Notes

Proofs

  1. ^ If denotes the zero vector in while denote the zero scalar, then absolute homogeneity implies that
  2. ^ 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 ).
  3. ^ Let and It remains to show that The triangle inequality implies Since as desired.

References

  1. ^ a b c d Kubrusly 2011, p. 200.
  2. ^ a b c Narici & Beckenstein 2011, pp. 120–121.
  3. ^ a b c d e f g h i j Narici & Beckenstein 2011, pp. 116–128.
  4. ^ a b c d e f g Wilansky 2013, pp. 15–21.
  5. ^ a b c d Schaefer & Wolff 1999, p. 40.
  6. ^ a b c d e f g Narici & Beckenstein 2011, pp. 177–220.
  7. ^ Narici & Beckenstein 2011, pp. 116−128.
  8. ^ Narici & Beckenstein 2011, pp. 107–113.
  9. ^ Schechter 1996, p. 691.
  10. ^ a b Narici & Beckenstein 2011, p. 149.
  11. ^ a b c d Narici & Beckenstein 2011, pp. 149–153.
  12. ^ a b c Wilansky 2013, pp. 18–21.
  13. ^ Obvious if is a real vector space. For the non-trivial direction, assume that on and let Let and be real numbers such that Then
  14. ^ Wilansky 2013, p. 20.
  15. ^ a b c d e f Wilansky 2013, pp. 21–26.
  16. ^ Narici & Beckenstein 2011, pp. 150.
  17. ^ Wilansky 2013, pp. 50–51.
  18. ^ a b c Narici & Beckenstein 2011, pp. 156–175.
  19. ^ a b Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
  20. ^ Wilansky 2013, pp. 49–50.
  21. ^ Narici & Beckenstein 2011, pp. 115–154.