Seminorm

In mathematics, particularly in functional analysis, a seminorm is a real-valued function on a vector space over the field of real or complex numbers that generalizes the notion of a norm by removing the positive definite requirement of a norm. This means that unlike a norm, a seminorm is allowed to potentially map non-zero vectors to zero. A seminormed space is a pair consisting of a vector space and a seminorm on it. Every norm is a seminorm and every normed space is a seminormed space, but there are seminorms that are not norms.

Seminorms are intimately connected with convex sets and locally convex topological vector spaces. In fact, a topological vector space (TVS) is locally convex if and only if its topology is induced by a family of seminorms.

Definition

Throughout, X will be a vector space over the field 𝕂, where 𝕂 is either the real numbers ℝ or the complex numbers ℂ. A scalar is any element in X's field 𝕂.

Seminorms

Definition: A map p : X → ℝ is called a seminorm if it satisfies the following two conditions:

  1. Subadditivity: p(x + y) ≤ p(x) + p(y) for all x, y ∈ X;
    • The above inequality is called the triangle inequality.
    • p is called additive if equality holds (i.e. if p(x + y) = p(x) + p(y) for all x, y ∈ X).
  2. Absolute homogeneity: p(sx) = |s| p(x) for all x ∈ X and all scalars s;

Definition: A seminorm p : X → ℝ is called a norm if in addition it satisfies the following condition:

  1. Positive definiteness/Separates points: If x ∈ X is such that p(x) = 0, then x = 0;

Every seminorm (and thus also every norm) has the following properties:

  1. Nonnegativity: p ≥ 0 (i.e. p(x) ≥ 0 for all x ∈ X).
    • Note that this condition is also often called positivity even though the requirement is still p ≥ 0 (rather than p(x) > 0 for all non-0 x ∈ X)
  2. Positive homogeneity: p(rx) = r p(x) for all x ∈ X and all positive real r > 0;
    • Observe that if p is absolutely homogeneous then it must also be positively homogeneous.

Sublinear functions

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. Thus, any result that is known about sublinear functions can be immediately applied to seminorms.

Definition: A map p : X → ℝ 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.

Ultraseminorm

Definition: A map p : X → ℝ is called an ultraseminorm or a non-Archimedean seminorm if it is a seminorm and satisfies the following additional condition:

  1. p(x + y) ≤ max {p(x), p(y)} for all x, y ∈ X.
Quasi-seminorm

Definition: A map p : X → ℝ is called a quasi-seminorm if it is absolutely homogeneous and satisfies the following additional condition:

  1. There exists some b ≤ 1 such that p(x + y) ≤ b(p(x) + p(y)) for all x, y ∈ X,

where 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. Note that the definition of a quasi-seminorm is the almost the same as the definition of a seminorm, except that subadditivity is replaced by a weaker condition.

k-seminorms

Definition: A map p : X → ℝ is called a k-seminorm if it is subadditive and satisfies the following additional condition:

  1. 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. Note that the definition of a quasi-seminorm is the almost the same as the definition of a seminorm, except that absolute homogeneity is replaced by a weaker condition.

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 < log2
2 b then there exists k-seminorm p on X equivalent to q.

Induced topology and pseudometric
Definition:[1] If p is a seminorm on X then the map dp : X × X → ℝ defined by dp(x, y) := p(x - y) is a translation-invariant pseudometric on X called the canonical pseudometric induced by p.

The canonical pseudometric dp is a metric if and only if p is a norm.

Definition: If p is a seminorm on X then the topology on X induced by the canonical pseudometric dp is called the p-topology or the topology induced by p.

The p-topology on X induced by a seminorm p makes X into a locally convex pseudometrizable topological vector space (TVS). This topology is Hausdorff if and only if p is a norm. This topology make X into a complete TVS if and only if the canonical pseudometric dp is a complete pseudometric.

Definition: If p is a seminorm on X and r is a real number, then the open p-ball of radius r at the origin in X is the set { x ∈ X : p(x) < r }. The closed p-ball of radius r at the origin in X is the set { x ∈ X : p(x) ≤ r }. The open (resp. closed) unit p-ball in X is the open (resp. closed) p-ball of radius 1 in X.

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.

Definition: A topological vector space X is called seminormable (resp. normable) if its topology is induced by some seminorm (resp. norm).

Stronger, weaker, and equivalent seminorms
Notation and definition: The phrase "f ≤ g on S" has its usual meaning (see footnote for details).[2]
Definition: 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 are satisfies:
  1. The topology on X induced by q is finer than the topology induced by p.
  2. There exists a real K > 0 such that p ≤ Kq on X.[1]
  3. p is bounded on {{math|{ x ∈ X : q(x) < 1 }.[1]
  4. Either p = 0 or else p ≠ 0 and {{{1}}}}.[1]
  5. If (xi)∞
    i=1     is a sequence in X then q(xi) → 0 implies p(xi) → 0.[1]
  6. If (pi)i ∈ I is a net in X then q(xi) → 0 implies p(xi) → 0.
Definition: We say that two seminorms p and q on X are equivalent if they satisfy any of the following conditions:
  1. The topology on X induced by q is the same as the topology induced by p.
  2. q is stronger than p and p is stronger than q.[1]
  3. If (xi)∞
    i=1     is a sequence in X then q(xi) → 0 if and only if p(xi) → 0.
  4. There exist positive real numbers r > 0 and R > 0 such that rq ≤ p ≤ Rq.

Seminormed spaces
Defintion: A seminormed space is a pair (X, p) considering of a vector space X and a seminorm p on X. If the seminorm p is also a norm then we call the seminormed space (X, p) a normed space.
Notation: If we say that some vector space X is a normed space but don't assign a symbol to its norm, then it should be assumed that ||⋅|| denotes its norm.
Definition:[3] If F : (X, p) → (Y, q) is a map between seminormed spaces then let ||F||p,q := sup { q(F(x)) : p(x) ≤ 1 }.

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

  1. F is continuous;
  2. ||F||p,q < ∞;[3]
  3. There exists a real K ≥ 0 such that p ≤ Kq;[3]
    • In this case, ||F||p,q ≤ K.

If F is continuous then q(F(x)) ≤ ||F||p,q p(x) for all x ∈ X.[3]

The space of all continuous linear maps F : (X, p) → (Y, q) between seminormed spaces is itself a seminormed space under the seminorm ||F||p,q. This seminorm is a norm if q is a norm.[3]

Hahn-Banach theorem:[3] 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.

Characterizations

A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.[4] A locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[5]

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

  1. p is a seminorm.
  2. p is a convex F-seminorm.
    • In particular, every seminorm is an F-seminorm.
  3. p is a convex balanced G-seminorm.[6]
When seminorms are norms

If p is a seminorm on X then the following are equivalent:

  1. p is a norm;
  2. for all x ∈ X, p(x) = 0 if and only if x = 0;
  3. {x ∈ X : p(x) < 1 } does not contain a non-trivial vector subspace.[7]
  4. there exists a Hausdorff TVS topology on X in which {x ∈ X : p(x) < 1 } is bounded.
When sublinear functions are linear

If p is a sublinear function on a real vector space X then the following are equivalent:[8]

  1. p is a linear functional;
  2. for every x ∈ X, p(x) + p(-x) ≤ 0;
  3. for every x ∈ X, p(x) + p(-x) = 0;

Examples
Definition:[9] If f is a real-valued sublinear function on X, then the map p(x) := max { f(x), f(-x)} defines a seminorm on X called the seminorm associated with f.
  • All norms are seminorms and all seminorms are sublinear functions. The converse, however, is not true in general.
  • Every vector space admits a norm: If x• = (xi)i ∈ I is a Hamel basis for a vector space X then the real-valued map that sends x = ∑i ∈ I sixi ∈ X (where all but finitely many of the scalars si are 0) to ∑i ∈ I |si| is a paranorm on X is a norm on X.[10]
  • If p and q are seminorms on X then so is x ↦ max { p(x), q(q) }.[11]
  • 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 }. Moreover, p∧q ≤ p and p∧q ≤ q.[1]
  • Any finite sum of seminorms is a seminorm.
  • If f is a linear functional on X then the map {{math|x ↦ |f(x)|, denoted by |f|, is a seminorm on X. If X is a TVS then f is continuous if and only if |f| is.
  • Every non-negative real scalar multiple of p is a seminorm.[11]
  • The trivial seminorm is the map on X that is identically 0 (i.e. p(x) = 0 for all x ∈ X). This seminorm induces the indiscrete topology on X.
  • Every linear form f on a vector space defines a seminorm by x → |f(x)|.

Minkowski functionals and seminorms
Main article: Minkowski functional

Seminorms on a vector space X are intimately tied, via Minkowski functionals, to subsets of X that are convex, balanced, and absorbing in the following way. 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 equal to p.

Thus, seminorms on X can be identified with subsets of X. This is beneficial because there are things that we can do with subsets of X (e.g. assign a topology) that we cannot do with seminorms.

When Minkowski functionals are seminorms

Let X be a vector space and let D be an absorbing disk in X (recall that a disk is just a convex and balanced set).

Definition: The gauge or Minkowski functional of D (or associated with D, or induced by D) is the real-valued function pD : X → [0, ∞) defined by
pD(x) := inf { r > 0 : x ∈ rD}.

Theorem — If D is an absorbing disk in a vector space X then the Minkowski functional of D, which is the map pD : X → [0, ∞) defined by

pD(x) := inf { r > 0 : x ∈ rD},

is a seminorm on X.

Relationship with seminorms

If p is a seminorm on X and D ⊆ X is a set satisfying

{ x ∈ X : p(x) < 1 } ⊆ D ⊆ { x ∈ X : p(x) ≤ 1 }

then D is absorbing in X and p = pD, where pD is the Minkowski functional associated with D (i.e. the guage of D).[12]

  • In particular, if D is as above and q is any seminorm on X, then q = p if and only if { x ∈ X : q(x) < 1 } ⊆ D ⊆ { x ∈ X : q(x) ≤ 1 }.[12]

Properties of Minkowski functionals

Let X be a real or complex vector space and let D be an absorbing disk in X.

  1. pD is a seminorm on X.
  2. For any scalar s ≠ 0, psD = 1/|s|pD.[13]
  3. If J is an absorbing disk in X and J ⊆ D then pD ≤ pJ
  4. pD is a norm on X if and only if D does not contain a non-trivial vector subspace.[7]

Poperties

Algebraic properties

Let X be a vector space over 𝔽 where 𝔽 is either the real or complex numbers.

Properties of sublinear functions

Since every seminorm is a sublinear function, seminorms have all of the following properties:

If p : X → [0, ∞) be a real-valued sublinear function on X then:

  • Every sublinear function is a convex functional.
  • p(0) = 0.
  • 0 ≤ max {p(x), p(-x) } and p(x) - p(y) ≤ p(x - y) for all x, y ∈ X.[14][8]
  • If p is a sublinear function on a real vector space X then there exists a linear functional f on X such that f ≤ p.[8]
  • If X is a real vector space, f is a linear functional on X, and p is a sublinear function on X, then f ≤ p on X if and only if f -1(1) ∩ { x ∈ X : p(x) < 1 } = ∅.[8]
Properties of seminorms

Theorem[15][16] (Extending seminorms) — If M is a vector subspace of X, p is a seminorm on M, and r 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. (see footnote for proof)[17]

If p : X → [0, ∞) is a seminorm on X then:

  • The second triangle inequality: |p(x) − p(y)| ≤ p(x − y) for all x, y ∈ X.
  • For all r > 0, {x ∈ X : p(x) < r } is an absorbing disk in X.[11]
  • p is a norm on X if and only if { x ∈ X : p(x) < 1 } does not contain a non-trivial vector subspace.
  • Every seminorm is a non-negative sublinear function. However, sublinear functions are not necessarily non-negative.
  • p -1(0) is a vector subspace of X.
  • For any x ∈ X and r > 0,[18]
    x + { y ∈ X : p(y) < r } = { y ∈ X : p(x - y) < r}.
  • For any r > 0,[11]
    r { x ∈ X : p(x) < 1 } = { x ∈ X : p(x) < r } = { x ∈ X : 1/rp(x) < 1}.
  • If D is a set satisfying { x ∈ X : p(x) < 1 } ⊆ D ⊆ { x ∈ X : p(x) ≤ 1 } then D is absorbing in X and p = pD, where pD is the Minkowski functional associated with D (i.e. the guage of D).[12]
    • In particular, if D is as above and q is any seminorm on X, then q = p if and only if { x ∈ X : q(x) < 1 } ⊆ D ⊆ { x ∈ X : q(x) ≤ 1 }.[12]

Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.

Inequalities involving seminorms

If p : X → [0, ∞) is a seminorm on X then:

  • If f is a linear functional on a real or complex vector space X and if p is a seminorm on X, then |f| ≤ p on X if and only if Re f ≤ p on X (see footnote for proof).[19][20]
  • If q is a seminorm on X, then p ≤ q if and only if q(x) ≤ 1 implies p(x) ≤ 1.[21]
  • If q is a seminorm on X and a > 0 and b > 0 are such that p(x) < a implies q(x) ≤ b, then aq(x) ≤ bp(x) for all x ∈ X. [16]
  • If f is a linear functional on X and 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. [16]
  • If X is a vector space over the reals and f is a non-0 linear functional on X, then f ≤ p if and only if ∅ = f -1(1) ∩ { x ∈ X : p(x) < 1 }.[21]
  • Suppose a and b are positive real numbers and q, p1, ..., pn are seminorms on X. If for every x ∈ X, pi(x) < a implies q(x) < b for all i, then aq ≤ b Σn
    i=1     pi    .[7]

Topological properties
  • If X is a TVS and p is a continuous seminorm on X, then the closure of { x ∈ X : p(x) < r } in X is equal to { x ∈ X : p(x) ≤ r }.[11]
  • 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 ∈ 𝒫 p -1(0).[22]
  • A subset S in a seminormed space (X, p) is bounded if and only if p(S) is bounded.[23]
Pseudometrizability
  • If (X, p) is a seminormed space then the locally convex topology that p induces on X makes X into a pseudometrizable TVS with a canonical pseudometric given by d(x, y) := p(x - y) for all x, y ∈ X.[13]

Continuity
See also: Continuous linear operator
Continuity and sublinear functions

Suppose p is a sublinear function on a TVS X. Then the following are equivalent:

  1. p is continuous;
  2. p is continuous at 0;
  3. p is uniformly continuous on X;

and if p is non-negative (i.e. p ≥ 0) then we may add to this list:

  1. { x ∈ X : p(x) < 1 } is open in X.

If X is a real TVS, f is a linear functional on X, and p is a continuous sublinear function on X, then f ≤ p on X implies that f is continuous.[8]

Continuity of seminorms

If p is a seminorm on a topological vector space X, then the following are equivalent:[12]

  1. p is continuous.
  2. p is continuous at 0;[11]
  3. is open in X;[11]
  4. is closed neighborhood of 0 in X;[11]
  5. p is uniformly continuous on X;[11]
  6. There exists a continuous seminorm q on X such that p ≤ q.[11]

In particular, if (X, p) is a seminormed space then a seminorm q on X is continuous if and only if q is dominated by a positive scalar multiple of p.[11]

Normability

A topological vector space (TVS) is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm). Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.

If X is a Hausdorff locally convex TVS then the following are equivalent:

  1. X is normable.
  2. X has a bounded neighborhood of the origin.
  3. the strong dual of X is normable.[24]
  4. the strong dual of X is metrizable.[24]

Furthermore, X 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).[25]

Generalizations

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.

gollark: It's easy, too, what with the 2-letter symbols.
gollark: _will now do this_
gollark: I wonder if anyone collects periodic table codes.
gollark: _ponders accepting incuhatchable egg_
gollark: I got eggs! Just what I always wanted.

See also

References
  1. Wilansky 2013, pp. 15-21.
  2. If f and g are functions and S is a set, then "f ≤ g on S" means that f(s) ≤ g(s) for every s ∈ S that belongs to both of their domains. If we don't specify S then S should be taken to be the set of all points that their domains share.
  3. Wilansky 2013, pp. 21-26.
  4. Wilansky 2013, pp. 50-51.
  5. Narici 2011, pp. 156-175.
  6. Schechter 1996, p. 691.
  7. Narici 2011, p. 149.
  8. Narici 2011, pp. 177-220.
  9. Narici 2011, pp. 120–121.
  10. Wilansky 2013, pp. 20-21.
  11. Narici 2011, pp. 116–128.
  12. Schaefer 1999, p. 40.
  13. Narici 2011, pp. 115-154.
  14. Narici 2011, pp. 120-121.
  15. Narici 2011, pp. 150.
  16. Wilansky 2013, pp. 18-21.
  17. Let S be the convex hull of { m ∈ M : p(x) ≤ 1 } ∪ { x ∈ X : q(x) ≤ 1 }. Note that S is an absorbing disk in X so let q be the Minkowski functional of S. Then p = P on M and P ≤ q on X.
  18. Narici 2011, pp. 116−128.
  19. Obvious if X is a real vector space. For the non-trivial direction, assume that Re f ≤ p on X and let x ∈ X. Let r ≥ 0 and t be real numbers such that f(x) = reit. Then |f(x)| = r = f(e-itx) = Re (f(e-itx)) ≤ p(e-itx) = p(x).
  20. Wilansky 2013, p. 20.
  21. Narici 2011, pp. 149–153.
  22. Narici 2011, pp. 149-153.
  23. Wilansky 2013, pp. 49-50.
  24. Treves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
  25. Narici 2011, pp. 156–175.
  • 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.CS1 maint: ref=harv (link)
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
  • 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.CS1 maint: ref=harv (link)
  • Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
  • Khaleelulla, S. M. (July 1, 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • 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.CS1 maint: ref=harv (link)
  • Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
  • Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 0-12-566060-X.CS1 maint: ref=harv (link)
  • 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.CS1 maint: ref=harv (link)
  • Schechter, Eric (October 30, 1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.CS1 maint: date and year (link)
  • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.CS1 maint: ref=harv (link)
  • Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
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