Linear form
The linear form is a term from the mathematical branch of linear algebra . It is used to describe a linear mapping from a vector space into the underlying body .
In the context of functional analysis , i.e. in the case of a topological - or - vector space, the linear forms considered are mostly continuous linear functionals .
definition
Let it be a body and a vector space. A mapping is called linear form if the following applies to all vectors and scalars :
- (Additivity);
- (Homogeneity).
The set of all linear forms over a given vector space forms its dual space and thus itself again naturally a vector space.
properties
General properties for linear forms are for example:
- Like any linear map, they are completely determined by their values for an arbitrary basis of .
- They are either trivial (identical everywhere ) or surjective .
- If two of them have the same kernel , the only difference is that they are multiplied by a scalar.
The following also applies specifically to linear functionals:
- They are continuous exactly when their core is closed .
- Their absolute amount is always a semi-norm on .
- Linear functionals are exactly the mappings , where a vector and denote the standard scalar product.
Linear form as a tensor
A linear form is a first order covariant tensor ; they are therefore sometimes also called 1-form . 1-forms form the basis for the introduction of differential forms .
Related terms
Is specific and to change the second condition in starting, whereby the complex conjugate of designated, to obtain a semi-linear form .
A mapping that is linear or semilinear in more than one argument is a sesquilinear form , a bilinear form , or generally a multilinear form .
literature
- Linear form . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Gerd Fischer : Linear Algebra . An introduction for first-year students. 16th, revised and expanded edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0428-0 , p. 280-281 .
- Harro Heuser : Functional Analysis . Theory and application (= mathematical guidelines ). 4th edition. BG Teubner , Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
- Eberhard Oeljeklaus, Reinhold Remmert : Lineare Algebra I (= Heidelberg pocket books . Volume 150 ). Springer Verlag, Berlin ( inter alia ) 1974, ISBN 3-540-06715-9 ( MR0366944 ).
- Walter Rudin : Functional Analysis , 2nd Ed., McGraw-Hill Inc., New York, 1991
- Dirk Werner : Functional Analysis . 5th, exp. Edition. Springer-Verlag Berlin Heidelberg, Berlin 2005, ISBN 3-540-21381-3 .