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 . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

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 : ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle f\colon V\to K}$ ${\displaystyle x,y\in V}$ ${\displaystyle \alpha \in K}$

1. ${\displaystyle f(x+y)=f(x)+f(y)}$ (Additivity);
2. ${\displaystyle f(\alpha x)=\alpha f(x)}$ (Homogeneity).

The set of all linear forms over a given vector space forms its dual space and thus itself again naturally a vector space. ${\displaystyle V}$ ${\displaystyle V^{*}}$ ${\displaystyle K}$

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 . ${\displaystyle V}$
• They are either trivial (identical everywhere ) or surjective . ${\displaystyle 0_{K}}$
• 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 . ${\displaystyle V}$
• Linear functionals are exactly the mappings , where a vector and denote the standard scalar product. ${\displaystyle \mathbb {K} ^{n}\to \mathbb {K} }$ ${\displaystyle x\mapsto \langle v,x\rangle }$ ${\displaystyle v\in \mathbb {K} ^{n}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

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 . ${\displaystyle f}$

Related terms

Is specific and to change the second condition in starting, whereby the complex conjugate of designated, to obtain a semi-linear form . ${\displaystyle K=\mathbb {C} }$ ${\displaystyle f(\alpha x)={\overline {\alpha }}f(x)}$ ${\displaystyle {\overline {\alpha }}}$ ${\displaystyle \alpha }$

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 .