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 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 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:

  1. Subadditivity/Triangle inequality: 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;

A consequence of these two properties is nonnegativity: p(x) ≥ 0 for all x ∈ X, which is equivalent to positive homogeneity: p(rx) = r p(x) for all x ∈ X and all positive real r > 0;

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

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

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. 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.

Pseudometrics and the induced topology

A seminorm p on X induces a topology via the translation-invariant pseudometric dp : X × X → ℝ; dp(x, y) = p(x - y). This topology is Hausdorff if and only if dp 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 seminorms

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:

  1. The topology on X induced by q is finer than the topology induced by p.
  2. If (xi)∞
    i=1     is a sequence in X, then q(xi) → 0 implies p(xi) → 0.[1]
  3. If (xi)i ∈ I is a net in X, then q(xi) → 0 implies p(xi) → 0.
  4. p is bounded on { x ∈ X : q(x) < 1 }.[1]
  5. If inf { q(x) : p(x) = 1, x ∈ X} = 0, then p(x) = 0 for all x.[1]
  6. 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. This happens 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.

Continuity
Continuity of seminorms

If p is a seminorm on a topological vector space X, then the following are equivalent:[2]

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

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.[3]

If X is a real TVS, f is a linear functional on X, and p is a continuous seminorm (or more generally, a sublinear function) on X, then f ≤ p on X implies that f is continuous.[4]

Continuity of linear maps

If F : (X, p) → (Y, q) is a map between seminormed spaces then let

||F||p,q := sup { q(F(x)) : p(x) ≤ 1  }.[5]

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 < ∞;[5]
  3. There exists a real K ≥ 0 such that p ≤ Kq;[5]
    • In this case, ||F||p,q ≤ K.

If F is continuous then q(F(x)) ≤ ||F||p,q p(x) for all x ∈ X.[5]

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.[5]

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 }.[3]
  • 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).[6]
  • A subset S in a seminormed space (X, p) is bounded if and only if p(S) is bounded.[7]
  • 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.[8]
  • 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).[6]
  • A subset S in a seminormed space (X, p) is (von Neumann) bounded if and only if p(S) is bounded.[7]
  • 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).[9]

Normability
See also: Normed space

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.[10]
  4. the strong dual of X is metrizable.[10]

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).[9]

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. 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.[11]

Examples
  • 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)}.[12]
  • 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)}.[3]
  • 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.

Algebraic properties

Let X be a vector space over 𝔽 where 𝔽 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 p : X → [0, ∞) be a real-valued sublinear function on X then:

  • Seminorms satisfy the reverse triangle inequality: |p(x) − p(y)| ≤ p(x − y) for all x, y ∈ X.[13][4]
  • For any x ∈ X and r > 0,[14] 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.[3]
  • 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.[13][4]
  • 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.[4]
  • 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 } = ∅.[4]
Other properties of seminorms

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

  • p is a norm on X if and only if { x ∈ X : p(x) < 1 } does not contain a non-trivial vector subspace.
  • p −1(0) is a vector subspace of X.
  • For any r > 0,[3]
    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 gauge of D).[2]
    • 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 }.[2]
  • If (X, ||⋅||) is a normed space then ||x - y|| = ||x - z|| + ||z - y|| for all x, y, z ∈ X.[15]
  • 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 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.[5]

A similar extension property also holds for seminorms:

Theorem[16][17] (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)[18]

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. [17]
  • 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 } = ∅.[4][21]
  • 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. [17]
  • 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    .[22]

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.[23] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[24]

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.
  3. p is a convex balanced G-seminorm.[25]

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

  1. p is a norm;
  2. {x ∈ X : p(x) < 1 } does not contain a non-trivial vector subspace.[22]
  3. There exists a normed on X, with respect to which, {x ∈ X : p(x) < 1 } is bounded.

If p is a sublinear function on a real vector space X then the following are equivalent:[4]

  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;

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.

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.

Weakening subadditivity: Quasi-seminorms

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 all x, 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-seminorms

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

See also

References
  1. Wilansky 2013, pp. 15-21.
  2. Schaefer 1999, p. 40.
  3. Narici & Beckenstein 2011, pp. 116–128.
  4. Narici & Beckenstein 2011, pp. 177-220.
  5. Wilansky 2013, pp. 21-26.
  6. Narici & Beckenstein 2011, pp. 149-153.
  7. Wilansky 2013, pp. 49-50.
  8. Narici & Beckenstein 2011, pp. 115-154.
  9. Narici & Beckenstein 2011, pp. 156–175.
  10. Treves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
  11. Schaefer & Wolff 1999, p. 40.
  12. Narici & Beckenstein 2011, pp. 120–121.
  13. Narici & Beckenstein 2011, pp. 120-121.
  14. Narici & Beckenstein 2011, pp. 116−128.
  15. Narici & Beckenstein 2011, pp. 107-113.
  16. Narici & Beckenstein 2011, pp. 150.
  17. Wilansky 2013, pp. 18-21.
  18. 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.
  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 & Beckenstein 2011, pp. 149–153.
  22. Narici & Beckenstein 2011, p. 149.
  23. Wilansky 2013, pp. 50-51.
  24. Narici & Beckenstein 2011, pp. 156-175.
  25. 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) [1981]. 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 (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.
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