# Semi-norm

A **semi-norm** or **seminorm** is a function in mathematics that is absolutely homogeneous and subadditive . It generalizes the concept of the norm by dispensing with the property of positive definiteness . Every semi-norm is nonnegative , symmetrical with respect to sign reversal , sublinear and convex . An associated standard can be derived from each semi-standard by forming residual classes . With the help of families of semi-norms, locally convex vector spaces can also be defined. Semi-norms are studied in particular in linear algebra and functional analysis .

## definition

Let be a vector space over the body . A seminorm on is a picture with the properties absolute homogeneity and subadditivity , that is for all and all are

- (absolute homogeneity)

and

- (Subadditivity),

where represents the magnitude of the scalar. A vector space together with a semi-standard is called a semi-normalized space .

## Examples

- Every norm is a semi-norm, which is also positively definite .
- The null function , which maps every element of the vector space to zero, is a semi-norm.
- The absolute value of a real or complex-valued linear function is a semi-norm.
- Every positive semidefinite symmetrical bilinear form in the complex case Hermitian sesquilinear form induced by a semi-norm. It is used here that the Cauchy-Schwarz inequality applies to every positive semidefinite symmetrical bilinear form (or Hermitian sesquilinear form), from which the subadditivity can be deduced.
- Is a topological space and compact , then by a half-norm on the space of all continuous functions given. It is used here that continuous functions are restricted to compact sets and therefore the supremum remains finite.
- The Minkowski functional for an absorbing , absolutely convex subset of a vector space.
- On the dual space of a normalized space defined for and a semi-norm.
- On the set of restricted linear operators , ( ) and ( ) can be used to define semi-norms.

## properties

By setting in the definition it follows immediately

- ,

the semi-norm of the zero vector is thus zero. In contrast to norms, however, there can also be vectors whose semi-norm is. By setting of , the subadditivity (also called triangle inequality ) and the absolute homogeneity result in nonnegativity

for everyone . By setting you can see that a semi-norm is symmetrical with respect to the sign reversal , that is

and from the application of the triangle inequality to the inverted triangle inequality follows

- .

Furthermore, a semi-norm is sublinear , since absolute homogeneity implies positive homogeneity , and also convex , because it applies to real things

- .

Conversely, every absolutely homogeneous and convex function is subadditive and therefore a semi-norm, which can be seen by setting and multiplying with .

## Residual class formation

Due to the absolute homogeneity and the subadditivity, the amount is

of vectors with semi-norm zero a subspace of . Therefore, an equivalence relation on by

To be defined. The vector space of all equivalence classes from the above equivalence relation is, together with the semi-norm, a normalized space . This process is called residual class formation in relation to the semi-norm and referred to as a factor space . This construction is used, for example, to define the *L ** ^{p}* spaces .

^{}## Family of semi-norms

In the functional analysis in the area of locally convex vector spaces , families of semi-norms are mostly considered. With these it can be possible to define a topology on the original vector space that makes it a topological vector space . To do this, one specifies that the set is open if there are one and a finite number of indices , so that

applies to all .

In this context, families with a certain separation characteristic are of particular interest. A family of semi-norms is called separating if there is at least one semi-norm for each such that it holds. A vector space is Hausdorffian with respect to the topology explained above if and only if the family of semi-norms is separating. Such a topological vector space is called a locally convex vector space.

## literature

- Walter Rudin : Functional Analysis . McGraw-Hill, New York 1991, ISBN 0-07-054236-8 .

## Individual evidence

- ^ Walter Rudin: Functional Analysis . McGraw-Hill, New York 1991, pp. 24-25 .
- ^ Walter Rudin: Functional Analysis . McGraw-Hill, New York 1991, pp. 26-27 .

## Web links

- EA Gorin: Semi-norm . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Todd Rowland:
*Seminorm*. In:*MathWorld*(English). - Robert Milson, D. Allan Drummond:
*Seminorm*. In:*PlanetMath .*(English)