Semi-norm

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

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 ${\displaystyle x-y+y}$

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

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 ${\displaystyle V}$ ${\displaystyle V}$

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

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

literature

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

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 .

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)