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 .
Let it be a body and a vector space. A mapping is called linear form if the following applies to all vectors and scalars :
The set of all linear forms over a given vector space forms its dual space and thus itself again naturally a vector space.
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 .
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 .
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