Seminorm

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: • Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. • 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 and that every seminorm p also has the following property: 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: 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 \rightarrow \mathbb{R} for which \lVert f \rVert_c := \left( \int_{\Omega}| f |^c \, d\mu \right)^{1/c} exists and is finite. It can be shown that X is a vector space, and the functional \lVert \cdot \rVert_c is a seminorm on X. However, it is not always a norm (e.g. if \Omega = \mathbb{R} and \mu is the Lebesgue measure) because \lVert h \rVert_c = 0 does not always imply h = 0. To make \lVert \cdot \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 \cdot \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 \times Y \to \R defined by r(x, y) = p(x) + q(y) is a seminorm (respectively, a norm) on X \times Y. In particular, the maps on X \times Y defined by (x, y) \mapsto p(x) and (x, y) \mapsto q(y) are both seminorms on X \times Y. If p and q are seminorms on X then so are (p \vee q)(x) = \max \{p(x), q(x)\} and (p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\} 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 \vee q) \wedge (p \vee r)) (x, y) = \inf \{\max(2|x_1|, |y_1|) + \max(|x_2|, 2|y_2|) : x = x_1 + x_2 \text{ and } y = y_1 + y_2\} 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\big\vert_{L(X)} is a norm on L(X). ==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\{x \in X : p(x) 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)| \leq p(x - y) and also 0 \leq \max \{p(x), p(-x)\} and p(x) - p(y) \leq p(x - y). For any vector x \in X and positive real r > 0: x + \{y \in X : p(y) and furthermore, \{x \in X : p(x) 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) 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) does not contain a non-trivial vector subspace. If p : X \to [0, \infty) 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). Furthermore, for any real r > 0, r \{x \in X : p(x) If D is a set satisfying \{x \in X : p(x) 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) If (X, \|\,\cdot\,\|) 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: p is a seminorm. p is a convex F-seminorm. p is a convex balanced G-seminorm. If any of the above conditions hold, then the following are equivalent: p is a norm; \{x \in X : p(x) does not contain a non-trivial vector subspace. There exists a norm on X, with respect to which, \{x \in X : p(x) is bounded. If p is a sublinear function on a real vector space X then the following are equivalent: p is a linear functional; p(x) + p(-x) \leq 0 \text{ for every } x \in X; p(x) + p(-x) = 0 \text{ for every } x \in X; Inequalities involving seminorms If p, q : X \to [0, \infty) 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) 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)\} then q(x) Then a q \leq b \left(p_1 + \cdots + 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) 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). f \leq p on X if and only if f^{-1}(1) \cap \{x \in X : p(x) If a > 0 and b > 0 are such that p(x) implies f(x) \neq 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 \times 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: \{x \in X : p(x) 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) 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: • The topology on X induced by q is finer than the topology induced by p. • If x_{\bull} = \left(x_i\right)_{i=1}^{\infty} is a sequence in X, then q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0 in \R implies p\left(x_{\bull}\right) \to 0 in \R. • If x_{\bull} = \left(x_i\right)_{i \in I} is a net in X, then q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0 in \R implies p\left(x_{\bull}\right) \to 0 in \R. • p is bounded on \{x \in X : q(x) • If \inf{} \{q(x) : p(x) = 1, x \in X\} = 0 then p(x) = 0 for all x \in X. • 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: The topology on X induced by q is the same as the topology induced by p. q is stronger than p and p is stronger than q. If x_{\bull} = \left(x_i\right)_{i=1}^{\infty} is a sequence in X then q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0 if and only if p\left(x_{\bull}\right) \to 0. There exist positive real numbers r > 0 and R > 0 such that r q \leq p \leq R q. Normability and seminormability 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: X is normable. X is seminormable. X has a bounded neighborhood of the origin. The strong dual X^{\prime}_b of X is normable. 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) 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 \mathcal{P} is equal to \bigcap_{p \in \mathcal{P}} p^{-1}(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: p is continuous. p is continuous at 0; \{x \in X : p(x) is open in X; \{x \in X : p(x) \leq 1\} is closed neighborhood of 0 in X; p is uniformly continuous on X; 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 \|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}. If F : (X, p) \to (Y, q) is a linear map between seminormed spaces then the following are equivalent: F is continuous; \|F\|_{p,q} ; 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)\} \text{ for all } 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)) \text{ for all } 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 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: {{block indent | em = 1.5 | text = Suppose that q is a quasi-seminorm on a vector space X with multiplier b. If 0 then there exists k-seminorm p on X equivalent to q.}} ==See also==