# Semi-norm

The function is a semi-norm in space${\ displaystyle p (x, y) = | xy |}$${\ displaystyle \ mathbb {R} ^ {2}}$

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 ${\ displaystyle V}$ ${\ displaystyle \ mathbb {K} \ in \ {\ mathbb {R}, \ mathbb {C} \}}$${\ displaystyle V}$${\ displaystyle p \ colon V \ to \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ lambda \ in \ mathbb {K}}$${\ displaystyle x, \, y \ in V}$

${\ displaystyle p (\ lambda x) = | \ lambda | p (x)}$   (absolute homogeneity)

and

${\ displaystyle p (x + y) \ leq p (x) + p (y)}$   (Subadditivity),

where represents the magnitude of the scalar. A vector space together with a semi-standard is called a semi-normalized space . ${\ displaystyle | \ cdot |}$ ${\ displaystyle (V, p)}$

## 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.${\ displaystyle p \ equiv 0}$
• 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.${\ displaystyle (\ cdot, \ cdot)}$${\ displaystyle p (x) = {\ sqrt {(x, x)}}}$
• Is a topological space and compact , then by a half-norm on the space of all continuous functions given. It is used here that continuous functions are restricted to compact sets and therefore the supremum remains finite.${\ displaystyle X}$${\ displaystyle K \ subset X}$ ${\ displaystyle \ textstyle p_ {K} (f): = \ sup _ {x \ in K} | f (x) |}$${\ displaystyle X \ rightarrow \ mathbb {C}}$
• The Minkowski functional for an absorbing , absolutely convex subset of a vector space.${\ displaystyle p_ {U}}$${\ displaystyle U}$
• On the dual space of a normalized space defined for and a semi-norm.${\ displaystyle X ^ {*}}$${\ displaystyle p_ {x} (\ varphi) = | \ varphi (x) |}$${\ displaystyle x \ in X}$${\ displaystyle \ varphi \ in X ^ {*}}$
• On the set of restricted linear operators , ( ) and ( ) can be used to define semi-norms.${\ displaystyle {\ mathfrak {L}} (X, Y)}$${\ displaystyle p_ {x} (T) = \ | Tx \ |}$${\ displaystyle x \ in X}$${\ displaystyle p_ {x, \ psi} (T) = | \ psi (Tx) |}$${\ displaystyle x \ in X, \ psi \ in Y ^ {*}}$

## properties

By setting in the definition it follows immediately ${\ displaystyle \ lambda = 0}$

${\ displaystyle p (0) = 0}$,

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${\ displaystyle x \ neq 0}$${\ displaystyle p (x) = 0}$${\ displaystyle y = -x}$

${\ displaystyle p (x) \ geq 0}$

for everyone . By setting you can see that a semi-norm is symmetrical with respect to the sign reversal , that is ${\ displaystyle x \ in V}$${\ displaystyle \ lambda = -1}$

${\ displaystyle p (x) = p (-x)}$

and from the application of the triangle inequality to the inverted triangle inequality follows${\ displaystyle x-y + y}$

${\ displaystyle | p (x) -p (y) | \ leq p (xy)}$.

Furthermore, a semi-norm is sublinear , since absolute homogeneity implies positive homogeneity , and also convex , because it applies to real things${\ displaystyle 0 \ leq t \ leq 1}$

${\ displaystyle p (tx + (1-t) y) \ leq p (tx) + p ((1-t) y) = tp (x) + (1-t) p (y)}$.

Conversely, every absolutely homogeneous and convex function is subadditive and therefore a semi-norm, which can be seen by setting and multiplying with . ${\ displaystyle t = {\ tfrac {1} {2}}}$${\ displaystyle 2}$

## Residual class formation

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

${\ displaystyle Z = \ {x \ in V \ colon p (x) = 0 \}}$

of vectors with semi-norm zero a subspace of . Therefore, an equivalence relation on by ${\ displaystyle V}$${\ displaystyle V}$

${\ displaystyle x \ sim y: \ Longleftrightarrow xy \ in Z}$

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 . ${\ displaystyle {\ tilde {V}}}$${\ displaystyle p}$${\ displaystyle V}$${\ displaystyle {\ tilde {V}}}$ ${\ displaystyle V / Z}$

## 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 ${\ displaystyle (p_ {i}) _ {i \ in I}}$${\ displaystyle V}$${\ displaystyle U \ subset V}$${\ displaystyle x \ in U}$${\ displaystyle \ epsilon> 0}$${\ displaystyle i_ {1}, \ ldots, i_ {r}}$

${\ displaystyle p_ {i_ {j}} (y) <\ epsilon, \, j = 1, \ ldots, r \ Rightarrow x + y \ in U}$

applies to all . ${\ displaystyle y \ in V}$

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. ${\ displaystyle (p_ {i}) _ {i \ in I}}$${\ displaystyle x \ in V \ setminus \ {0 \}}$${\ displaystyle p_ {i}}$${\ displaystyle p_ {i} (x) \ neq 0}$${\ displaystyle V}$

## Individual evidence

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