Seminorm

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 $$x, y \in X.$$
 * 2) 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 $$p(x) := \max \{f(x), f(-x)\}.$$</li> <li>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).</li> <li>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 $$X \times Y.$$</li> <li>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 $$p \wedge q \leq q.$$ </li> <li>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 $$(p \vee q \wedge r) (x, y) := \max(|x|, |y|)$$</li> <li>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 $$L(X).$$</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$$\{x \in X : p(x) < 1\}$$ 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: $$|p(x) - p(y)| \leq p(x - y)$$ 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 $$r > 0:$$ $$x + \{y \in X : p(y) < r\} = \{y \in X : p(x - y) < r\}$$ 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 $$p : X \to [0, \infty)$$ 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 $$r > 0,$$ $$r \{x \in X : p(x) < 1\} = \{x \in X : p(x) < r\} = \left\{x \in X : \tfrac{1}{r} p(x) < 1 \right\}.$$

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 $$(X, \|\,\cdot\,\|)$$ 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: <ol> <li>$$p$$ is a seminorm.</li> <li>$$p$$ is a convex $F$-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>$$\{x \in X : p(x) < 1\}$$ does not contain a non-trivial vector subspace.</li> <li>There exists a norm on $$X,$$ with respect to which, $$\{x \in X : p(x) < 1\}$$ 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;</li> <li>$$p(x) + p(-x) \leq 0 \text{ for every } x \in X$$;</li> <li>$$p(x) + p(-x) = 0 \text{ for every } x \in X$$;</li> </ol>

Inequalities involving seminorms
If $$p, q : X \to [0, \infty)$$ are seminorms on $$X$$ then: <ul> <li>$$p \leq q$$ if and only if $$q(x) \leq 1$$ implies $$p(x) \leq 1.$$</li> <li>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 $$x \in X.$$ </li> <li>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 $$a q \leq b \left(p_1 + \cdots + p_n\right).$$</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 $$\varnothing = f^{-1}(1) \cap \{x \in X : p(x) < 1\}.$$</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 $$\operatorname{Re} f \leq p$$ on $$X$$ (see footnote for proof). </li> <li>$$f \leq p$$ on $$X$$ if and only if $$f^{-1}(1) \cap \{x \in X : p(x) < 1 = \varnothing\}.$$</li> <li>If $$a > 0$$ and $$b > 0$$ are such that $$p(x) < a$$ implies $$f(x) \neq b,$$ then $$a |f(x)| \leq b p(x)$$ for all $$x \in X.$$</li> </ul>

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 $$X.$$ $$\blacksquare$$

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: $$\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}$$ 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 $$p.$$
 * 2) 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 $$\R.$$
 * 3) 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 $$\R.$$
 * 4) $$p$$ is bounded on $$\{x \in X : q(x) < 1\}.$$
 * 5) If $$\inf{} \{q(x) : p(x) = 1, x \in X\} = 0$$ then $$p(x) = 0$$ for all $$x \in X.$$
 * 6) 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: <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 $$x_{\bull} = \left(x_i\right)_{i=1}^{\infty}$$ 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 $$p\left(x_{\bull}\right) \to 0.$$</li> <li>There exist positive real numbers $$r > 0$$ and $$R > 0$$ such that $$r q \leq p \leq R q.$$</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 $$X^{\prime}_b$$ of $$X$$ is normable.</li> <li>The strong dual $$X^{\prime}_b$$ of $$X$$ is metrizable.</li> </ol> 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
<ul> <li>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 $$\{x \in X : p(x) \leq r\}.$$</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 $$\bigcap_{p \in \mathcal{P}} p^{-1}(0).$$</li> <li>A subset $$S$$ in a seminormed space $$(X, p)$$ is bounded if and only if $$p(S)$$ is bounded.</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 with a canonical pseudometric given by $$d(x, y) := p(x - y)$$ 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>$$\{x \in X : p(x) < 1\}$$ is open in $$X$$;</li> <li>$$\{x \in X : p(x) \leq 1\}$$ 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 $$F : (X, p) \to (Y, q)$$ 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: <ol> <li>$$F$$ is continuous;</li> <li>$$\|F\|_{p,q} < \infty$$;</li> <li>There exists a real $$K \geq 0$$ such that $$p \leq K q$$; </ol> If $$F$$ is continuous then $$q(F(x)) \leq \|F\|_{p,q} p(x)$$ for all $$x \in X.$$
 * In this case, $$\|F\|_{p,q} \leq K.$$</li>

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 $$0 < k \leq 1$$ 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: