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 seminorm 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. 
By definition, a norm on formula_1 is a seminorm that also separates points, meaning that it has the following additional property:
A seminormed space is a pair formula_13 consisting of a vector space formula_1 and a seminorm formula_11 on formula_16 If the seminorm is also a norm then we call the seminormed space formula_13 a normed space
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map formula_4 is called a sublinear function if it is subadditive (i.e. condition 1 above) and positively homogeneous (i.e. condition 5 above). 
Unlike a seminorm, a sublinear function is "not" necessarily nonnegative. 
Sublinear functions are often encountered in the context of the Hahn-Banach theorem.
Pseudometrics and the induced topology.
A seminorm on induces a topology via the translation-invariant pseudometric ; . This topology is Hausdorff if and only if is a metric, which occurs if and only if is a norm.
Equivalently, every vector space "V" with seminorm "p" induces a vector space quotient , where "W" is the subspace of "V" consisting of all vectors with . carries a norm defined by . The resulting topology, pulled back to , is precisely the topology induced by .
Any seminorm-induced topology makes locally convex, as follows. If is a seminorm on and is a real number, call the set } the open ball of radius about the origin; likewise the closed ball of radius is }. The set of all open (resp. closed) -balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the -topology on .
Stronger, weaker, and equivalent seminorms.
The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If and are seminorms on , then we say that is stronger than and that is weaker than if any of the following equivalent conditions holds:
 and are equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:
Continuity.
If is a seminorm on a topological vector space , then the following are equivalent: 
In particular, if is a seminormed space then a seminorm on is continuous if and only if is dominated by a positive scalar multiple of .
If is a real TVS, is a linear functional on , and is a continuous seminorm (or more generally, a sublinear function) on , then on implies that is continuous.
If is a map between seminormed spaces then let
If is a linear map between seminormed spaces then the following are equivalent:
If is continuous then for all .
The space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm . 
This seminorm is a norm if is a norm.
Topological properties.
Normability.
Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.
If is a Hausdorff locally convex TVS then the following are equivalent: 
Furthermore, is finite dimensional if and only if formula_19 is normable (here formula_19 denotes formula_21 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 (i.e. 0-dimensional).
Minkowski functionals and seminorms.
Seminorms on a vector space are intimately tied, via Minkowski functionals, to subsets of that are convex, balanced, and absorbing. Given such a subset of , the Minkowski functional of is a seminorm. Conversely, given a seminorm on , 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 .
Algebraic properties.
Let be a vector space over where is either the real or complex numbers.
Since every seminorm is a sublinear function, seminorms have all of the following properties:
If be a real-valued sublinear function on then:
If is a seminorm on 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:
Inequalities involving seminorms.
If is a seminorm on then:
Relationship to other norm-like concepts.
A topological vector space 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.
Let be a non-negative function. The following are equivalent:
If any of the above conditions hold, then the following are equivalent:
If is a sublinear function on a real vector space then the following are equivalent: 
Generalizations.
The concept of norm in composition algebras does "not" share the usual properties of a norm.
A composition algebra consists of an algebra over a field , an involution *, and a quadratic form , which is called the "norm". In several cases is an isotropic quadratic form so that has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.
An ultraseminorm or a non-Archimedean seminorm is a seminorm that also satisfies } for all .
Weakening subadditivity: Quasi-seminorms.
A map is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some such that for all .The smallest value of for which this holds is called the multiplier of .
A quasi-seminorm that separates points is called a quasi-norm on .
Weakening homogeneity: -seminorms.
A map is called a -seminorm if it is subadditive and there exists a such that and for all and scalars ,A -seminorm that separates points is called a -norm on .
We have the following relationship between quasi-seminorms and -seminorms: