# Seminorm

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 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.

## DefinitionEdit

Let X be a vector space over either the real numbers ℝ or the complex numbers ℂ. A map *p* : *X* → ℝ is called a **seminorm** if it satisfies the following two conditions:

- Subadditivity/Triangle inequality:
*p*(*x*+*y*) ≤*p*(*x*) +*p*(*y*) for all*x*,*y*∈*X*; - Homogeneity:
*p*(*sx*) = |*s*|*p*(*x*) for all*x*∈*X*and all scalars s;

A consequence of these two properties is nonnegativity: *p*(*x*) ≥ 0 for all *x* ∈ *X*.

Note that a seminorm is also a norm if (and only if) it also separates points: *p*(*x*) = 0 implies *x* = 0.

## Pseudometrics and the induced topologyEdit

A seminorm p on X induces a topology via the translation-invariant pseudometric *d*_{p} : *X* × *X* → ℝ; *d*_{p}(*x*, *y*) = *p*(*x* - *y*). This topology is Hausdorff if and only if *d*_{p} is a metric, which occurs if and only if p is a norm.^{[1]}

Equivalently, every vector space *V* with seminorm *p* induces a vector space quotient *V*/*W*, where *W* is the subspace of *V* consisting of all vectors **v** ∈ *V* with *p*(**v**) = 0. *V*/*W* carries a norm defined by *p*(**v**+*W*) = *p*(**v**). The resulting topology, pulled back to V, is precisely the topology induced by p.

Any seminorm-induced topology makes X locally convex, as follows. If p is a seminorm on X and r is a real number, call the set { *x* ∈ *X* : *p*(*x*) < *r* } the open ball of radius r about the origin; likewise the closed ball of radius r is { *x* ∈ *X* : *p*(*x*) ≤ *r* }. The set of all open (resp. closed) p-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the p-topology on X.

### Stronger, weaker, and equivalent seminormsEdit

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If p and q are seminorms on X, then we say that q is **stronger** than p and that p is **weaker** than q if any of the following equivalent conditions holds:

- The topology on X induced by q is finer than the topology induced by p.
- If (
*x*_{i})^{∞}_{i=1}is a sequence in X, then*q*(*x*_{i}) → 0 implies*p*(*x*_{i}) → 0.^{[1]} - If (
*x*_{i})_{i ∈ I}is a net in X, then*q*(*x*_{i}) → 0 implies*p*(*x*_{i}) → 0. - p is bounded on {
*x*∈*X*:*q*(*x*) < 1 }.^{[1]} - If inf {
*q*(*x*) :*p*(*x*) = 1,*x*∈*X*} = 0, then*p(x)*= 0 for all x.^{[1]} - There exists a real
*K*> 0 such that*p*≤*Kq*on X.^{[1]}

p and q are **equivalent** if they are both weaker (or both stronger) than each other.

### Hom spaceEdit

If *F* : (*X*, *p*) → (*Y*, *q*) is a linear map between seminormed spaces then the following are equivalent:

- F is continuous;
- ||
*F*||_{p→q}= sup {*q*(*F*(*x*)) :*p*(*x*) ≤ 1} < ∞.^{[2]}

||*F*||_{p→q} is itself a seminorm on the space of all continuous linear maps *F* : (*X*, *p*) → (*Y*, *q*), and a full norm if and only if q is.^{[2]}

## Examples and elementary propertiesEdit

- The
**trivial seminorm**on X (*p*(*x*) = 0 for all*x*∈*X*) induces the indiscrete topology on X. - Every linear form
*f*on a vector space defines a seminorm by*x*→ |*f*(*x*)|. - Every real-valued sublinear function f on X defines a seminorm
*p*(*x*) = max {*f*(*x*),*f*(-*x*)}.^{[3]} - Any finite sum of seminorms is a seminorm.
- If p and q are seminorms on X then so is (
*p*∨*q*)(*x*) = max {*p*(*x*),*q*(*x*)}.^{[4]} - If p and q are seminorms on X then so is (
*p*∧*q*)(*x*) := inf {*p*(*y*) +*q*(*z*) :*x*=*y*+*z*with*y*,*z*∈*X*}. *p*∧*q*≤*p*and*p*∧*q*≤*q*.^{[1]}- Moreover, the space of seminorms on X is a distributive lattice with respect to the above operations.

### Convexity and boundednessEdit

- Seminorms satisfy the reverse triangle inequality: |
*p*(*x*) −*p*(*y*)| ≤*p*(*x*−*y*) for all*x*,*y*∈*X*.^{[5]}^{[6]} - For any
*x*∈*X*and*r*> 0,^{[7]}*x*+ {*y*∈*X*:*p*(*y*) <*r*} = {*y*∈*X*:*p*(*x*-*y*) <*r*}.

- Since every seminorm is a sublinear function, every seminorm p on X is a convex function. Moreover, for all
*r*> 0, {*x*∈*X*:*p*(*x*) <*r*} is an absorbing disk in X.^{[4]} - If f is a linear functional on X, then
*f*≤*p*on X if and only if*f*^{-1}(1) ∩ {*x*∈*X*:*p*(*x*) < 1 } = ∅.^{[6]}^{[8]}More generally, if*a*> 0 and*b*> 0 are such that*p*(*x*) <*a*implies*f*(*x*) ≠*b*, then*a*|*f*(*x*)| ≤*bp*(*x*) for all*x*∈*X*.^{[9]} - Seminorms offer a particularly clean formulation of the Hahn-Banach theorem: If M is a vector subspace of a seminormed space (
*X*,*p*) and if f is a continuous linear functional on M, then f may be extended to a continuous linear functional F on X that has the same norm as f.^{[2]}- A similar extension property also holds for seminorms: if q is a seminorm on X such that
*p*≤*q*|_{M}, then there exists a seminorm P on X such that*P*|_{M}=*p*and*P*≤*q*. To see this, let S be the convex hull of {*m*∈*M*:*p*(*x*) ≤ 1 } ∪ {*x*∈*X*:*q*(*x*) ≤ 1 }. S is an absorbing disk; its Minkowski functional is the desired extension.^{[10]}^{[9]}

- A similar extension property also holds for seminorms: if q is a seminorm on X such that

### Seminormed spacesEdit

- The closure of { 0 } in a locally convex space X whose topology is defined by a family of continuous seminorms 𝒫 is equal to
*p*^{-1}(0).^{[11]} - A subset S in a seminormed space (
*X*,*p*) is (von Neumann) bounded if and only if*p*(*S*) is bounded.^{[12]} - The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces trivial (i.e. 0-dimensional).
^{[13]}

## Minkowski functionals and seminormsEdit

Seminorms on a vector space X are intimately tied, via Minkowski functionals, to subsets of X that are convex, balanced, and absorbing. Given such a subset D of X, the Minkowski functional of D is a seminorm. Conversely, given a seminorm p on X, the sets { *x* ∈ *X* : *p*(*x*) < 1 } and { *x* ∈ *X* : *p*(*x*) ≤ 1 } 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 p.^{[14]}

## Relationship to other norm-like conceptsEdit

A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin.^{[15]} Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.^{[16]}

Let *p* : *X* → ℝ be a non-negative function. The following are equivalent:

- p is a seminorm.
- p is a convex F-seminorm.
- p is a convex balanced G-seminorm.
^{[17]}

If any of the above conditions hold, then the following are equivalent:

- p is a norm;
- {
*x*∈*X*:*p*(*x*) < 1 } does not contain a non-trivial vector subspace.^{[18]} - There exists a normed on X, with respect to which, {
*x*∈*X*:*p*(*x*) < 1 } is bounded.

An **ultraseminorm** or a **non-Archimedean seminorm** is a seminorm *p* : *X* → ℝ that also satisfies *p*(*x* + *y*) ≤ max {*p*(*x*), *p*(*y*)} for all *x*, *y* ∈ *X*.

### GeneralizationsEdit

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

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

#### Weakening subadditivity: Quasi-seminormsEdit

A map *p* : *X* → ℝ is called a **quasi-seminorm** if it is (absolutely) homogeneous and there exists some *b* ≤ 1 such that

p(x+y) ≤b(p(x) +p(y)) for allx,y∈X.

The smallest value of b for which this holds is called the **multiplier of p**.

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

#### Weakening homogeneity: k-seminormsEdit

A map *p* : *X* → ℝ is called a **k-seminorm** if it is subadditive and there exists a k such that 0 < *k* ≤ 1 and for all *x* ∈ *X* and scalars s,

p(sx) = |s|^{k}p(x)

A k-seminorm that separates points is called a **k-norm** on X.

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

- Suppose that q is a quasi-seminorm on a vector space X with multiplier b. If 0 < √
*k*< log_{2}*b*then there exists k-seminorm p on X equivalent to q.

## See alsoEdit

- 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

## ReferencesEdit

- ^
^{a}^{b}^{c}^{d}^{e}^{f}Wilansky 2013, pp. 15-21. - ^
^{a}^{b}^{c}Wilansky 2013, pp. 21-26. **^**Narici & Beckenstein 2011, pp. 120–121.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 116–128. **^**Narici & Beckenstein 2011, pp. 120-121.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 177-220. **^**Narici & Beckenstein 2011, pp. 116−128.**^**Narici & Beckenstein 2011, pp. 149–153.- ^
^{a}^{b}Wilansky 2013, pp. 18-21. **^**Narici & Beckenstein 2011, pp. 150.**^**Narici & Beckenstein 2011, pp. 149-153.**^**Wilansky 2013, pp. 49-50.**^**Narici & Beckenstein 2011, pp. 156–175.**^**Schaefer & Wolff 1999, p. 40.**^**Wilansky 2013, pp. 50-51.**^**Narici & Beckenstein 2011, pp. 156-175.**^**Schechter 1996, p. 691.**^**Narici & Beckenstein 2011, p. 149.

- Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
*Topological Vector Spaces: The Theory Without Convexity Conditions*. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. - Jarchow, Hans (1981).
*Locally convex spaces*. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. - Bourbaki, Nicolas (1987) [1981].
*Topological Vector Spaces: Chapters 1–5*[*Sur certains espaces vectoriels topologiques*].*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). Written at Berlin Heidelberg.
*Counterexamples in Topological Vector Spaces*. Lecture Notes in Mathematics.**936**. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. - Köthe, Gottfried (1969).
*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) [1967].
*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.