In mathematics, particularly in functional analysis, a seminorm is like a norm but 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

be a vector space over either the real numbers or the complex A real-valued function is called a if it satisfies the following two conditions:

1. Subadditivity/Triangle inequality

2. Absolute homogeneity:

also has the following property: Nonnegativity:

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 is a seminorm that also separates points, meaning that it has the following additional property: Positive definite/Positive/

A is called a .

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function 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 is a seminorm if and only if it is a sublinear and balanced function.

Examples

The induces the indiscrete topology is the Lebesgue measure. The resulting space. is any linear form on a vector space then its absolute value is a seminorm. A sublinear function is a seminorm if and only if it is a Every real-valued sublinear function 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). then so are is generally not a distributive lattice is a linear map

Minkowski functionals and seminorms

that are convex, balanced, and absorbing

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity: the reverse triangle inequality:

is an absorbing disk

Other properties of seminorms

Every seminorm is a balanced function does not contain a non-trivial vector subspace.

is absorbing denotes the Minkowski functional).

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

be a non-negative function. The following are equivalent: is a seminorm. is a convex F-seminorm. is a convex balanced G-seminorm.

If any of the above conditions hold, then the following are equivalent: is a norm; does not contain a non-trivial vector subspace. There exists a norm is bounded.

then the following are equivalent: is a linear functional; ; ;

Inequalities involving seminorms

then:

then: (see footnote for proof).

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:

A similar extension property also holds for seminorms:

1. Proof be the convex hull is an absorbing disk and so the Minkowski functional

Topologies of seminormed spaces

Pseudometrics and the induced topology

induces a topology, called the , via the canonical translation-invariant pseudometric This topology is Hausdorff is a norm. into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis 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 .

induces a vector space quotient The resulting topology, pulled back

locally convex the -balls at the origin forms a neighborhood basis of convex balanced

Stronger, weaker, and equivalent seminorms

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms is is if any of the following equivalent conditions holds:

1. is a net

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 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.

is a Hausdorff locally convex TVS then the following are equivalent: is normable. is seminormable. has a bounded neighborhood of the origin. The strong dual is normable. is metrizable. 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

is bounded is bounded. into a pseudometrizable TVS 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

then the following are equivalent: is continuous. is continuous at 0; ; ; ;

is continuous.

Continuity of linear maps

is a map between seminormed spaces then let

is a linear map between seminormed spaces then the following are equivalent: is continuous; ; ;

is a norm.

Generalizations

The concept of in composition algebras does share the usual properties of a norm.

consists of an algebra over a field an involution and a quadratic form is an isotropic quadratic form has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An or a

Weakening subadditivity: Quasi-seminorms

is called a for which this holds is called the

A quasi-seminorm that separates points is called a

-seminorms

is called a -seminorm that separates points is called a

-seminorms:

See also

Notes

Proofs

References

External links

• Sublinear functions

• The sandwich theorem for sublinear and super linear functionals