Seminorm Explained

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

Let

X be a vector space over either the real numbers \R or the complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions:

1. Subadditivity/Triangle inequality:

p(x + y) \leq p(x) + p(y) for all x, y \in X.

1. Absolute homogeneity:

p(s x) = |s|p(x) for all x \in X and all scalars s.

These two conditions imply that

p( 0 ) = 0 ^[1] and that every seminorm p also has the following property:^[2]

3. Nonnegativity: p(x) \geq 0 for all x \in X.

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

X is a seminorm that also separates points, meaning that it has the following additional property:

4. Positive definite/Positive/: whenever x \in X satisfies p(x) = 0 , then x = 0 .

A is a pair

(X, p) consisting of a vector space X and a seminorm p on X. If the seminorm p is also a norm then the seminormed space (X, p) is called a .

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

p : X \to \R is called a if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. A real-valued function p : X \to \R is a seminorm if and only if it is a sublinear and balanced function.

Examples

• The on
  X, which refers to the constant 0 map on X, induces the indiscrete topology on X.
• Let \mu be a measure on a space \Omega . For an arbitrary constant c \geq 1 , let X be the set of all functions f: \Omega → R for which$$\backslash lVert\; f\; \backslash rVert\_c\; :=\; \backslash left(\backslash int\_|\; f\; |^c\; \backslash ,\; d\backslash mu\; \backslash right)^$$exists and is finite. It can be shown that X is a vector space, and the functional \lVert ⋅ \rVert _{c} is a seminorm on X . However, it is not always a norm (e.g. if \Omega = R and \mu is the Lebesgue measure) because \lVert h \rVert _{c} = 0 does not always imply h = 0 . To make \lVert ⋅ \rVert _{c} a norm, quotient X by the closed subspace of functions h with \lVert h \rVert _{c} = 0 . The resulting space, L ^{c(\mu)} , has a norm induced by \lVert ⋅ \rVert _{c} .
• If f is any linear form on a vector space then its absolute value |f|, defined by x \mapsto |f(x)|, is a seminorm.
• A sublinear function f : X \to \R on a real vector space X is a seminorm if and only if it is a, meaning that f( - x) = f(x) for all x \in X.
• Every real-valued sublinear function f : X \to \R on a real vector space X induces a seminorm p : X \to \R defined by p(x) : = max \{f(x), f( - x)\}.
• 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 p : X \to \R and q : Y \to \R are seminorms (respectively, norms) on X and Y then the map r : X x Y \to \R defined by r(x, y) = p(x) + q(y) is a seminorm (respectively, a norm) on X x Y. In particular, the maps on X x Y defined by (x, y) \mapsto p(x) and (x, y) \mapsto q(y) are both seminorms on X x Y.
• If p and q are seminorms on X then so are $$(p\; \backslash vee\; q)(x)\; =\; \backslash max\; \backslash$$ and $$(p\; \backslash wedge\; q)(x)\; :=\; \backslash inf\; \backslash$$where p \wedge q \leq p and p \wedge q \leq q.
• The space of seminorms on X is generally not a distributive lattice with respect to the above operations. For example, over \R ^{2} , p(x, y) : = max (|x|, |y|), q(x, y) : = 2 |x|, r(x, y) : = 2 |y| are such that $$((p\; \backslash vee\; q)\; \backslash wedge\; (p\; \backslash vee\; r))\; (x,\; y)\; =\; \backslash inf\; \backslash$$ while (p \vee q \wedge r) (x, y) : = max (|x|, |y|)
• If L : X \to Y is a linear map and q : Y \to \R is a seminorm on Y, then q \circ L : X \to \R is a seminorm on X. The seminorm q \circ L will be a norm on X if and only if L is injective and the restriction q \ vert _{L(X)} is a norm on L(X).

Minkowski functionals and seminorms

See 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 \in X : p(x) < 1 \} and \{x \in X : p(x) \leq 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.

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity,

p( 0 ) = 0 , and for all vectors x, y \in X :the reverse triangle inequality: $$|p(x)\; -\; p(y)|\; \backslash leq\; p(x\; -\; y)$$and also$0\; \backslash leq\; \backslash max\; \backslash$ and p(x) - p(y) \leq p(x - y).

For any vector

x \in X and positive real r > 0 : $$x\; +\; \backslash \; =\; \backslash$$and furthermore, \{x \in X : p(x) < r\} is an absorbing disk in X.

If

p is a sublinear function on a real vector space X then there exists a linear functional f on X such that f \leq p and furthermore, for any linear functional g on X, g \leq p on X if and only if g ^{-1} ( 1 ) \cap \{x \in X : p(x) < 1 \} = \varnothing.

Other properties of seminorms

Every seminorm is a balanced function. A seminorm

p is a norm on X if and only if \{x \in X : p(x) < 1 \} does not contain a non-trivial vector subspace.

If

p : X \to [ 0 , inf ty) is a seminorm on X then \ker p : = p ^{-1} ( 0 ) is a vector subspace of X and for every x \in X, p is constant on the set x + \ker p = \{x + k : p(k) = 0 \} and equal to p(x). ^[3]

Furthermore, for any real

r > 0 , $$r\; \backslash \; =\; \backslash \; =\; \backslash left\backslash .$$

If

D is a set satisfying \{x \in X : p(x) < 1 \} \subseteq D \subseteq \{x \in X : p(x) \leq 1 \} then D is absorbing in X and p = p _{D} where p _{D} denotes the Minkowski functional associated with D (that is, the gauge of D ). In particular, if D is as above and q is any seminorm on X, then q = p if and only if \{x \in X : q(x) < 1 \} \subseteq D \subseteq \{x \in X : q(x) \leq\}.

If

(X, \| ⋅ \|) is a normed space and x, y \in X then \|x - y\| = \|x - z\| + \|z - y\| for all z in the interval [x, y].

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

Let

p : X \to \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.

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

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

If

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

1. p is a linear functional;
2. p(x) + p( - x) \leq 0 forevery x \in X ;
3. p(x) + p( - x) = 0 forevery x \in X ;

Inequalities involving seminorms

If

p, q : X \to [ 0 , inf ty) are seminorms on X then:

• p \leq q if and only if q(x) \leq 1 implies p(x) \leq 1 .
• If a > 0 and b > 0 are such that p(x) < a implies q(x) \leq b, then a q(x) \leq b p(x) for all x \in X.
• Suppose a and b are positive real numbers and q, p _{1,} \ldots, p _{n} are seminorms on X such that for every x \in X, if max \{p _{1(x),} \ldots, p _{n(x)\}} < a then q(x) < b. Then a q \leq b \left(p _{1} + … + p _{n\right).}
• If X is a vector space over the reals and f is a non-zero linear functional on X, then f \leq p if and only if \varnothing = f ^{-1} ( 1 ) \cap \{x \in X : p(x) < 1 \}.

If

p is a seminorm on X and f is a linear functional on X then:

• |f| \leq p on X if and only if \operatorname{Re} f \leq p on X (see footnote for proof).^[4]
• f \leq p on X if and only if f ^{-1} ( 1 ) \cap \{x \in X : p(x) < 1 = \varnothing\}.
• If a > 0 and b > 0 are such that p(x) < a implies f(x) ≠ b, then a |f(x)| \leq b p(x) for all x \in X.

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.

A similar extension property also holds for seminorms:

Proof: Let

S be the convex hull of \{m \in M : p(m) \leq 1 \} \cup \{x \in X : q(x) \leq 1 \}. Then S is an absorbing disk in X and so the Minkowski functional P of S is a seminorm on X. This seminorm satisfies p = P on M and P \leq q on X. \blacksquare

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm

p on X induces a topology, called the, via the canonical translation-invariant pseudometric d _{p} : X x X \to \R ; d _{p(x,} y) : = p(x - y) = p(y - x). This topology is Hausdorff if and only if d _{p} is a metric, which occurs if and only if p is a norm. This topology makes X 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:$$\backslash \; \backslash quad\; \backslash text\; \backslash quad\; \backslash$$as r > 0 ranges over the positive reals. Every seminormed space (X, p) 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 .

Equivalently, every vector space

X with seminorm p induces a vector space quotient X / W, where W is the subspace of X consisting of all vectors x \in X with p(x) = 0 . Then X / W carries a norm defined by p(x + W) = p(x). The resulting topology, pulled back to X, 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 \in \R, call the set \{x \in X : p(x) < r\} the ; likewise the closed ball of radius r is \{x \in X : p(x) \leq 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 than p and that p is 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.

1. If

x _{\bull} = \left(x _{i\right)} ^{inf ty} ^{i =1} is a sequence in X, then q\left(x _{\bull} \right) : = \left(q\left(x _{i\right)\right)} ^{inf ty} ^{i =1} \to 0 in \R implies p\left(x _{\bull} \right) \to 0 in \R.

1. If

x _{\bull} = \left(x _{i\right)} _{i} is a net in X, then q\left(x _{\bull} \right) : = \left(q\left(x _{i\right)\right)} _{i} \to 0 in \R implies p\left(x _{\bull} \right) \to 0 in \R. p is bounded on \{x \in X : q(x) < 1 \}.

1. If

inf {} \{q(x) : p(x) = 1 , x \in X\} = 0 then p(x) = 0 for all x \in X.

1. There exists a real

K > 0 such that p \leq K q on X.

The seminorms

p and q are called 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.
3. If x _{\bull} = \left(x _{i\right)} ^{inf ty} ^{i =1} is a sequence in X then q\left(x _{\bull} \right) : = \left(q\left(x _{i\right)\right)} ^{inf ty} ^{i =1} \to 0 if and only if p\left(x _{\bull} \right) \to 0 .
4. There exist positive real numbers r > 0 and R > 0 such that r q \leq p \leq R q.

Normability and seminormability

See also: Normed space.

A topological vector space (TVS) is said to be a (respectively, a) 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 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. Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.A TVS is normable if and only if it is a T₁ space and admits a bounded convex neighborhood of the origin.

If

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

1. X is normable.
2. X is seminormable.
3. X has a bounded neighborhood of the origin.
4. The strong dual X ^{\prime} _{b} of X is normable.
5. The strong dual X ^{\prime} _{b} of X is metrizable.

Furthermore, X is finite dimensional if and only if X ^{\prime} _{\sigma} is normable (here X ^{\prime} _{\sigma} denotes X ^{\prime} 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).

Topological properties

• If
  X is a TVS and p is a continuous seminorm on X, then the closure of \{x \in X : p(x) < r\} in X is equal to \{x \in X : p(x) \leq r\}.
• The closure of \{ 0 \} in a locally convex space X whose topology is defined by a family of continuous seminorms l{P} is equal to c ap _{p} } p^(0).
• A subset S in a seminormed space (X, p) is bounded if and only if p(S) is bounded.
• 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 \in X.
• 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).

Continuity of seminorms

If

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

1. p is continuous.
2. p is continuous at 0;
3. \{x \in X : p(x) < 1 \} is open in X ;
4. \{x \in X : p(x) \leq 1 \} is closed neighborhood of 0 in X ;
5. p is uniformly continuous on X ;
6. There exists a continuous seminorm q on X such that p \leq q.

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.

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 \leq p on X implies that f is continuous.

Continuity of linear maps

If

F : (X, p) \to (Y, q) is a map between seminormed spaces then let$$\backslash |F\backslash |\_\; :=\; \backslash sup\; \backslash .$$

If

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

1. F is continuous;
2. \|F\| _{p,q} < inf ty ;
3. There exists a real K \geq 0 such that p \leq K q ;
   • In this case,
   \|F\| _{p,q} \leq K.

If F is continuous then q(F(x)) \leq \|F\| _{p,q} p(x) for all x \in X.

The space of all continuous linear maps

F : (X, p) \to (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.

Generalizations

The concept of in composition algebras does 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 or a is a seminorm

p : X \to \R that also satisfies p(x + y) \leq max \{p(x), p(y)\} forall x, y \in X.

Weakening subadditivity: Quasi-seminorms

A map

p : X \to \R is called a if it is (absolutely) homogeneous and there exists some b \leq 1 such that p(x + y) \leq b p(p(x) + p(y)) forall x, y \in X. The smallest value of b for which this holds is called the

A quasi-seminorm that separates points is called a on

X.

Weakening homogeneity -

k -seminorms

A map

p : X \to \R is called a if it is subadditive and there exists a k such that 0 < k \leq 1 and for all x \in X and scalars s, $$p(s\; x)\; =\; |s|^k\; p(x)$$ A k -seminorm that separates points is called a on X.

We have the following relationship between quasi-seminorms and

k -seminorms:

Notes

Proofs

References

• • • • • • • • • • • • Book: Prugovečki, Eduard. Quantum mechanics in Hilbert space. 1981. 2nd. Academic Press. 20. 0-12-566060-X.

External links

• Sublinear functions
• The sandwich theorem for sublinear and super linear functionals

Notes and References

1. If z \in X denotes the zero vector in X while 0 denote the zero scalar, then absolute homogeneity implies that p( 0 ) = p( 0 z) = | 0 |p(z) = 0 p(z) = 0 . \blacksquare
2. Suppose p : X \to \R is a seminorm and let x \in X. Then absolute homogeneity implies p( - x) = p(( -1 ) x) = | -1 |p(x) = p(x). The triangle inequality now implies p( 0 ) = p(x + ( - x)) \leq p(x) + p( - x) = p(x) + p(x) = 2 p(x). Because x was an arbitrary vector in X, it follows that p( 0 ) \leq 2 p( 0 ), which implies that 0 \leq p( 0 ) (by subtracting p( 0 ) from both sides). Thus 0 \leq p( 0 ) \leq 2 p(x) which implies 0 \leq p(x) (by multiplying thru by 1 / 2 ).
3. Let x \in X and k \in p ^{-1} ( 0 ). It remains to show that p(x + k) = p(x). The triangle inequality implies p(x + k) \leq p(x) + p(k) = p(x) + 0 = p(x). Since p( - k) = 0 , p(x) = p(x) - p( - k) \leq p(x - ( - k)) = p(x + k), as desired. \blacksquare
4. Obvious if X is a real vector space. For the non-trivial direction, assume that \operatorname{Re} f \leq p on X and let x \in X. Let r \geq 0 and t be real numbers such that f(x) = r e ^{i} . Then |f(x)| = r = f\left(e ^{-it} x\right) = \operatorname{Re}\left(f\left(e ^{-it} x\right)\right) \leq p\left(e ^{-it} x\right) = p(x).