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.

Definition

Let X be a vector space over either the real numbers or the complex numbers . 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, yX;
2. Homogeneity: p(sx) = |s| p(x) for all xX and all scalars s;

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

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 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 vV 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 { xX : p(x) < r } the open ball of radius r about the origin; likewise the closed ball of radius r is { xX : 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)iI is a net in X, then q(xi) → 0 implies p(xi) → 0.
4. p is bounded on { xX : q(x) < 1 }.[1]
5. If inf { q(x) : p(x) = 1, xX} = 0, then p(x) = 0 for all x.[1]
6. There exists a real K > 0 such that pKq on X.[1]

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

Hom space

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

1. F is continuous;
2. ||F||pq = sup { q(F(x)) : p(x) ≤ 1} < ∞.[2]

||F||pq 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 properties

• The trivial seminorm on X (p(x) = 0 for all xX) 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 (pq)(x) = max { p(x), q(x)}.[4]
• If p and q are seminorms on X then so is (pq)(x) := inf { p(y) + q(z) : x = y + z with y, zX}.
• pqp and pqq.[1]
• Moreover, the space of seminorms on X is a distributive lattice with respect to the above operations.

Convexity and boundedness

• Seminorms satisfy the reverse triangle inequality: |p(x) − p(y)| ≤ p(xy) for all x, yX.[5][6]
• For any xX and r > 0,[7] x + { yX : p(y) < r } = { yX : 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, {xX : p(x) < r} is an absorbing disk in X.[4]
• If f is a linear functional on X, then fp on X if and only if f-1(1) ∩ { xX : 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 xX. [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 pq|M, then there exists a seminorm P on X such that P|M = p and Pq. To see this, let S be the convex hull of { mM : p(x) ≤ 1 } ∪ { xX : q(x) ≤ 1 }. S is an absorbing disk; its Minkowski functional is the desired extension.[10][9]

Seminormed spaces

• 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 seminorms

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 { xX : p(x) < 1 } and { xX : 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 concepts

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:

1. p is a seminorm.
2. p is a convex F-seminorm.
3. p is a convex balanced G-seminorm.[17]

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

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

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.

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, yX.

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 xX 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.

References

1. Wilansky 2013, pp. 15-21.
2. ^ a b c Wilansky 2013, pp. 21-26.
3. ^ Narici & Beckenstein 2011, pp. 120–121.
4. ^ a b Narici & Beckenstein 2011, pp. 116–128.
5. ^ Narici & Beckenstein 2011, pp. 120-121.
6. ^ a b Narici & Beckenstein 2011, pp. 177-220.
7. ^ Narici & Beckenstein 2011, pp. 116−128.
8. ^ Narici & Beckenstein 2011, pp. 149–153.
9. ^ a b Wilansky 2013, pp. 18-21.
10. ^ Narici & Beckenstein 2011, pp. 150.
11. ^ Narici & Beckenstein 2011, pp. 149-153.
12. ^ Wilansky 2013, pp. 49-50.
13. ^ Narici & Beckenstein 2011, pp. 156–175.
14. ^ Schaefer & Wolff 1999, p. 40.
15. ^ Wilansky 2013, pp. 50-51.
16. ^ Narici & Beckenstein 2011, pp. 156-175.
17. ^ Schechter 1996, p. 691.
18. ^ Narici & Beckenstein 2011, p. 149.
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