In mathematics, particularly in functional analysis, a seminorm is a vector space 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.
- 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.
By definition, a norm on is a seminorm that also separates points, meaning that it has the following additional property:
- Positive definite/Point-separating: for all if then
A seminormed space is a pair consisting of a vector space and a seminorm on If the seminorm is also a norm then we call the seminormed space 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 (i.e. condition 1 above) and positively homogeneous (i.e. condition 5 above). Unlike a seminorm, a sublinear function is not necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn-Banach theorem.
Pseudometrics and the induced topology
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:
- The topology on induced by is finer than the topology induced by
- If is a sequence in then in implies in 
- If is a net in then in implies in
- is bounded on 
- If then for all 
- There exists a real such that on 
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:
- The topology on induced by is the same as the topology induced by
- is stronger than and is stronger than 
- If is a sequence in then if and only if
- There exist positive real numbers and such that
- Continuity of seminorms
If is a seminorm on a topological vector space then the following are equivalent:
- is continuous.
- is continuous at 0;
- is open in ;
- is closed neighborhood of 0 in ;
- is uniformly continuous on ;
- There exists a continuous seminorm on such that 
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 
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.
- Continuity of linear maps
If is a map between seminormed spaces then let
If is a linear map between seminormed spaces then the following are equivalent:
If is continuous then for all 
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.
- 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 (that is, 0-dimensional).
If is a Hausdorff locally convex TVS then the following are equivalent:
- is normable.
- has a bounded neighborhood of the origin.
- The strong dual of is normable.
- The strong dual of is metrizable.
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 (i.e. 0-dimensional).
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 
- The trivial seminorm on which refers to the constant map on induces the indiscrete topology on
- If is any linear form on a vector space then its absolute value defined by is a seminorm.
- Every real-valued sublinear function induces a seminorm on defined by 
- Any finite sum of seminorms is a seminorm.
- If and are seminorms on then so arewhere and 
- Moreover, the space of seminorms on is a distributive lattice with respect to the above operations.
Let be a vector space over where is either the real or complex numbers.
- Properties of seminorms because they are sublinear functions
Since every seminorm is a sublinear function, seminorms have all of the following properties:
If is a real-valued sublinear function on then:
- Seminorms satisfy the reverse triangle inequality: 
- For any and 
- Since every seminorm is a sublinear function, every seminorm on is a convex function. Moreover, for all is an absorbing disk in 
- Every sublinear function is a convex functional.
- and for all 
- 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
If is a seminorm on then:
- is a norm on if and only if does not contain a non-trivial vector subspace.
- is a vector subspace of
- For any 
- If is a set satisfying then is absorbing in and where denotes the Minkowski functional associated with (i.e. the gauge of ).
- 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 
- Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.
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 
A similar extension property also holds for seminorms:
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 (see footnote for proof).
- on if and only if 
- If and are such that implies then for all 
Relationship to other norm-like concepts
A topological vector space 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.
Let be a non-negative function. The following are equivalent:
If any of the above conditions hold, then the following are equivalent:
- is a norm;
- does not contain a non-trivial vector subspace.
- 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:
- is a linear functional;
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
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
- Asymmetric norm – Generalization of the concept of a norm
- Banach space – Normed vector space that is complete
- Hahn-Banach theorem
- Gowers norm
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Mahalanobis distance
- Matrix norm – Norm on a vector space of matrices
- Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
- Minkowski functional
- Norm (mathematics) – Length in a vector space
- Normed vector space – Vector space on which a distance is defined
- Relation of norms and metrics
- Sublinear function
- Topological vector space – Vector space with a notion of nearness
- If denotes the zero vector in while denote the zero scalar, then absolute homogeneity implies that
- 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 ).
- Wilansky 2013, pp. 15–21.
- Schaefer & Wolff 1999, p. 40.
- Narici & Beckenstein 2011, pp. 116–128.
- Narici & Beckenstein 2011, pp. 177–220.
- Wilansky 2013, pp. 21–26.
- Narici & Beckenstein 2011, pp. 149–153.
- Wilansky 2013, pp. 49–50.
- Narici & Beckenstein 2011, pp. 115–154.
- Narici & Beckenstein 2011, pp. 156–175.
- Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
- Narici & Beckenstein 2011, pp. 120–121.
- Narici & Beckenstein 2011, pp. 116−128.
- Narici & Beckenstein 2011, pp. 107–113.
- Let be the convex hull of Note that is an absorbing disk in so let be the Minkowski functional of Then on and on
- Narici & Beckenstein 2011, pp. 150.
- Wilansky 2013, pp. 18–21.
- Narici & Beckenstein 2011, p. 149.
- 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
- Wilansky 2013, p. 20.
- Wilansky 2013, pp. 50–51.
- Schechter 1996, p. 691.
- Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
- Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
- Bourbaki, Nicolas (1987) . Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
- Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
- Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
- Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Köthe, Gottfried (1983) . Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 0-12-566060-X.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
- Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
- Trèves, François (2006) . Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.