Antilinear

Als Semilinearform bezeichnet man in der linearen Algebra eine semilineare Abbildung aus einem Vektorraum in den diesem zugrundeliegenden Skalarkörper; relevant sind Semilinearformen lediglich über dem Körper C der komplexen Zahlen.

Eine Semilinearform f:V\longrightarrow K genügt den beiden folgenden Bedingungen, die ganz allgemein eine semilineare Abbildung kennzeichnen: für alle x, y aus V und alle λ aus K gilt:

  • (1) Superposition: f(x + y) = f(x) + f(y)
  • (2) komplex konjugierte Homogenität: f(\lambda x)=\overline{\lambda}f(x), wobei \overline{\lambda} das komplex Konjugierte von λ bezeichne.

Außer dem Begriff „semilinear“ finden sich in der Literatur auch die Begriffe „antilinear“ und „konjugiert linear“.

Siehe auch


Wikimedia Foundation.

Schlagen Sie auch in anderen Wörterbüchern nach:

  • antilinear — adjective Describing a particular mapping of a complex vector space …   Wiktionary

  • Antilinear map — In mathematics, a mapping from a complex vector space to another is said to be antilinear (or conjugate linear or semilinear, though the latter term is more general) if for all a, b in C and all x, y in V, where and are the complex conjugates of… …   Wikipedia

  • Complex conjugate vector space — In mathematics, the (formal) complex conjugate of a complex vector space is the complex vector space consisting of all formal complex conjugates of elements of . That is, is a vector space whose elements are in one to one correspondence with the… …   Wikipedia

  • Complex conjugate — Geometric representation of z and its conjugate in the complex plane In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magni …   Wikipedia

  • Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

  • Quaternionic representation — In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map:jcolon V o V, which satisfies:j^2=… …   Wikipedia

  • Real representation — In the mathematical field of representation theory a real representation is usually a representation on a real vector space U , but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an… …   Wikipedia

  • Tomita–Takesaki theory — In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution …   Wikipedia

  • Reality structure — In mathematics, particularly in representation theory, a reality structure on a complex vector space V of dimension n provides a means for identifying a real subspace V R of V so that V itself splits as a direct sum into real and imaginary parts …   Wikipedia

  • Bilinear — Im mathematischen Teilgebiet der linearen Algebra und verwandten Gebieten verallgemeinern die bilinearen Abbildungen die verschiedensten Begriffe von Produkten (im Sinne einer Multiplikation). Die Bilinearität entspricht dem Distributivgesetz bei …   Deutsch Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”