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 is called a if it satisfies the following two conditions: These two conditions imply that p(0) = 0 and that every seminorm p also has the following property:

<li>[Nonnegativity](https://bliptext.com/articles/nonnegative): p(x) \geq 0 for all x \in X.</li> </ol> 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](https://bliptext.com/articles/norm-mathematics) on X is a semi[norm](https://bliptext.com/articles/norm-mathematics) that also separates points, meaning that it has the following additional property: <li>[Positive definite](https://bliptext.com/articles/positive-definite)/Positive/: whenever x \in X satisfies p(x) = 0, then x = 0.</li> </ol> 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](https://bliptext.com/articles/sublinear-function). A map is called a if it is subadditive and [positive homogeneous](https://bliptext.com/articles/positive-homogeneous). Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem). A real-valued function is a seminorm if and only if it is a [sublinear](https://bliptext.com/articles/sublinear-function) and [balanced function](https://bliptext.com/articles/balanced-function).

Examples

<ul> <li>The on X, which refers to the constant 0 map on X, induces the [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) on X.</li> <li>Let \mu be a measure on a space \Omega. For an arbitrary constant c \geq 1, let X be the set of all functions for which exists and is finite. It can be shown that X is a vector space, and the functional is a seminorm on X. However, it is not always a norm (e.g. if and \mu is the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure)) because does not always imply h = 0. To make a norm, quotient X by the closed subspace of functions h with. The [resulting space](https://bliptext.com/articles/lp-space), L^c(\mu), has a norm induced by .</li> <li>If f is any [linear form](https://bliptext.com/articles/linear-form) on a vector space then its [absolute value](https://bliptext.com/articles/absolute-value) |f|, defined by is a seminorm.</li> <li>A [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X is a seminorm if and only if it is a, meaning that for all x \in X.</li> <li>Every real-valued [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X induces a seminorm defined by </li> <li>Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a [vector subspace](https://bliptext.com/articles/vector-subspace) is once again a seminorm (respectively, norm).</li> <li>If and are seminorms (respectively, norms) on X and Y then the map defined by is a seminorm (respectively, a norm) on X \times Y. In particular, the maps on X \times Y defined by and are both seminorms on X \times Y.</li> <li>If p and q are seminorms on X then so are and where and </li> <li>The space of seminorms on X is generally not a [distributive lattice](https://bliptext.com/articles/distributive-lattice) with respect to the above operations. For example, over \R^2, are such that while </li> <li>If L : X \to Y is a [linear map](https://bliptext.com/articles/linear-map) and is a seminorm on Y, then 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 is a norm on L(X).</li> </ul>

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 and 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: and also and For any vector x \in X and positive real r > 0: and furthermore, 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 Other properties of seminorms Every seminorm is a balanced function. A seminorm p is a norm on X if and only if does not contain a non-trivial vector subspace. If is a seminorm on X then is a vector subspace of X and for every x \in X, p is constant on the set and equal to p(x). Furthermore, for any real r > 0, If D is a set satisfying 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 If is a normed space and x, y \in X then 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 be a non-negative function. The following are equivalent:

<ol> <li>p is a seminorm.</li> <li>p is a [convex](https://bliptext.com/articles/convex-function) [F-seminorm](https://bliptext.com/articles/f-seminorm).</li> <li>p is a convex balanced G-seminorm.</li> </ol> If any of the above conditions hold, then the following are equivalent: <ol> <li>p is a norm;</li> <li> does not contain a non-trivial vector subspace.</li> <li>There exists a [norm](https://bliptext.com/articles/normed-vector-space) on X, with respect to which, is bounded.</li> </ol> If p is a sublinear function on a real vector space X then the following are equivalent: <ol> <li>p is a [linear functional](https://bliptext.com/articles/linear-functional);</li> <li>;</li> <li>;</li> </ol>

Inequalities involving seminorms

If are seminorms on X then:

<ul> <li>p \leq q if and only if q(x) \leq 1 implies </li> <li>If a > 0 and b > 0 are such that p(x) < a implies then for all x \in X. </li> <li>Suppose a and b are positive real numbers and are seminorms on X such that for every x \in X, if then q(x) < b. Then </li> <li>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 </li> </ul> If p is a seminorm on X and f is a linear functional on X then: <ul> <li>|f| \leq p on X if and only if on X (see footnote for proof). </li> <li>f \leq p on X if and only if </li> <li>If a > 0 and b > 0 are such that p(x) < a implies then for all x \in X.</li> </ul>

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem: A similar extension property also holds for seminorms:

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 ; 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: 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 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 the ; likewise the closed ball of radius r is 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 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:

<ol> <li>The topology on X induced by q is the same as the topology induced by p.</li> <li>q is stronger than p and p is stronger than q.</li> <li>If is a sequence in X then if and only if </li> <li>There exist positive real numbers r > 0 and R > 0 such that </li> </ol>

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:

<ol> <li>X is normable.</li> <li>X is seminormable.</li> <li>X has a bounded neighborhood of the origin.</li> <li>The [strong dual](https://bliptext.com/articles/strong-dual) of X is normable.</li> <li>The strong dual of X is [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space).</li> </ol> Furthermore, X is finite dimensional if and only if is normable (here denotes X^{\prime} endowed with the [weak-* topology](https://bliptext.com/articles/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

<ul> <li>If X is a TVS and p is a continuous seminorm on X, then the closure of in X is equal to </li> <li>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 </li> <li>A subset S in a seminormed space (X, p) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) if and only if p(S) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space).</li> <li>If (X, p) is a seminormed space then the locally convex topology that p induces on X makes X into a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space) with a canonical pseudometric given by for all x, y \in X.</li> <li>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).</li> </ul>

Continuity of seminorms

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

<ol> <li>p is continuous.</li> <li>p is continuous at 0;</li> <li> is open in X;</li> <li> is closed neighborhood of 0 in X;</li> <li>p is uniformly continuous on X;</li> <li>There exists a continuous seminorm q on X such that p \leq q.</li> </ol> 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 is a map between seminormed spaces then let If is a linear map between seminormed spaces then the following are equivalent:

<ol> <li>F is continuous;</li> <li>;</li> <li>There exists a real K \geq 0 such that p \leq K q; </ol> If F is continuous then for all x \in X. The space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm 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 that also satisfies Weakening subadditivity: Quasi-seminorms A map is called a if it is (absolutely) homogeneous and there exists some b \leq 1 such that 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 is called a if it is subadditive and there exists a k such that and for all x \in X and scalars s, A k-seminorm that separates points is called a on X. We have the following relationship between quasi-seminorms and k-seminorms: