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

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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

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  • The trivial seminorm 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 (e.g. if   and   is the Lebesgue measure) 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 symmetric function, meaning that   for all  
  • Every real-valued sublinear function   on a real vector space   induces a seminorm   defined by  [2]
  • Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace 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[3]   and   where   and  [4]
  • 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

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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

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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

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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

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If   are seminorms on   then:

  •   if and only if   implies  [11]
  • If   and   are such that   implies   then   for all   [12]
  • Suppose   and   are positive real numbers and   are seminorms on   such that for every   if   then   Then  [10]
  • If   is a vector space over the reals and   is a non-zero linear functional on   then   if and only if  [11]

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

  •   on   if and only if   on   (see footnote for proof).[13][14]
  •   on   if and only if  [6][11]
  • If   and   are such that   implies   then   for all  [12]

Hahn–Banach theorem for seminorms

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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

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Pseudometrics and the induced topology

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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

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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

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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

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  • If   is a TVS and   is a continuous seminorm on   then the closure of   in   is equal to  [3]
  • The closure of   in a locally convex space   whose topology is defined by a family of continuous seminorms   is equal to  [11]
  • A subset   in a seminormed space   is bounded if and only if   is bounded.[20]
  • 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  [21]
  • 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).[18]

Continuity of seminorms

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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

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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

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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

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Notes

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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

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  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.
  • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 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. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
  • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 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. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
  • Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: Birkhäuser. ISBN 978-0-8176-4998-2. OCLC 710154895.
  • 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. Vol. 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.
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  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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