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 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 formula_1 be a vector space over either the real numbers formula_2 or the complex numbers formula_3 A real-valued function formula_4 is called a if it satisfies the following  two conditions: These two conditions imply that formula_10 and that every seminorm formula_11 also has the following property: 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 formula_1 is a seminorm that also separates points, meaning that it has the following additional property: A is a pair formula_13 consisting of a vector space formula_1 and a seminorm formula_11 on formula_16 If the seminorm formula_11 is also a norm then the seminormed space formula_13 is  called a . Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map formula_4 is called a if it is subadditive (that is, condition 1 above) and positively homogeneous (that is, condition 5 above). Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. Minkowski functionals and seminorms. Seminorms on a vector space formula_1 are intimately tied, via Minkowski functionals, to subsets of formula_1 that are convex, balanced, and absorbing. Given such a subset formula_22 of formula_23 the Minkowski functional  of formula_22 is a seminorm. Conversely, given a seminorm formula_11 on formula_23 the setsformula_27 and formula_28 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 formula_29 Algebraic properties. Let formula_1 be a vector space over formula_31 where formula_31 is either the real or complex numbers. Properties of seminorms because they are sublinear functions Since every seminorm is a sublinear function, seminorms have all of the following properties: If formula_33 is a real-valued sublinear function on formula_1 then: Other properties of seminorms If formula_33  is a seminorm on formula_1 then: Relationship to other norm-like concepts. Let formula_4 be a non-negative function. The following are equivalent: If any of the above conditions hold, then the following are equivalent: If formula_11 is a sublinear function on a real vector space formula_1 then the following are equivalent: Inequalities involving seminorms. If formula_40 are seminorms on formula_1 then: If formula_11 is a seminorm on formula_1 and formula_44 is a linear functional on formula_1 then: 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 formula_11 on formula_1 induces a topology, called the , via the canonical translation-invariant pseudometric formula_68; formula_69 This topology is Hausdorff if and only if formula_70 is a metric, which occurs if and only if formula_11 is a norm. This topology makes formula_1 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: formula_73 as formula_74 ranges over the positive reals . Every seminormed space formula_13 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 formula_1 with seminorm formula_11 induces a vector space quotient formula_78 where formula_79 is the subspace of formula_1 consisting of all vectors formula_8 with formula_82 Then formula_83 carries a norm defined by formula_84 The resulting topology, pulled back to formula_23 is precisely the topology induced by formula_29 Any seminorm-induced topology makes formula_1 locally convex, as follows. If formula_11 is a seminorm on formula_1 and formula_90 call the  set formula_91 the ; likewise the closed ball of radius formula_92 is formula_93 The set of all open (resp. closed) formula_11-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the formula_11-topology on formula_16 Stronger, weaker, and equivalent seminorms. The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If formula_11 and formula_98 are seminorms on formula_23 then we say that formula_98 is than formula_11 and that formula_11 is than formula_98 if any of the following equivalent conditions holds: The seminorms formula_11 and formula_98  are called if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions: 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 T (because a TVS is Hausdorff if and only if it is a T space). A is a topological vector space that posses 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 T space and admits a bounded convex neighborhood of the origin. If formula_1 is a Hausdorff locally convex TVS then the following are equivalent: Furthermore, formula_1 is finite dimensional if and only if formula_131 is normable (here formula_131  denotes formula_133 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). Continuity of seminorms. If formula_11 is a seminorm on a topological vector space formula_23 then the following are equivalent: In particular, if formula_13 is a seminormed space then a seminorm formula_98 on formula_1 is continuous if and only if formula_98 is dominated by a positive scalar multiple of formula_29 If formula_1 is a real TVS, formula_44 is a linear functional on formula_23 and formula_11 is a continuous  seminorm (or more generally, a sublinear function) on formula_23 then formula_146 on formula_1 implies that formula_44 is continuous. Continuity of linear maps. If formula_149 is a map between seminormed spaces then let formula_150 If formula_149 is a linear map between seminormed spaces then the following are equivalent: If formula_51 is continuous then formula_153 for all formula_123 The space of all continuous linear maps formula_149 between seminormed spaces is itself a seminormed space under the seminorm formula_156 This seminorm is a norm if formula_98 is a norm. Generalizations. The concept of in composition algebras does share the usual properties of a  norm. A composition algebra formula_158 consists of an algebra over a field formula_159 an involution formula_160 and a quadratic form formula_161 which is called the "norm". In several cases formula_162 is an isotropic quadratic form so that formula_163 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 formula_4 that also satisfies formula_165 Weakening subadditivity: Quasi-seminorms A map formula_4 is called a if it is (absolutely) homogeneous and there exists some formula_167 such that formula_168 The smallest value of formula_169 for which  this holds is called the A quasi-seminorm that separates points is called a on formula_16 Weakening homogeneity - formula_171-seminorms A map formula_4 is called a if it is subadditive and there exists a formula_171 such that formula_174 and for all formula_8 and scalars formula_176formula_177A formula_171-seminorm that separates points is called a on formula_16 We have the following relationship between quasi-seminorms and formula_171-seminorms: Notes. Proofs