Linearform

The linear form is a term from the mathematical subfield of linear algebra. It refers to a linear map from a vector space to the underlying body.

In the context of functional analysis, that is, in the case of a topological -, or - vector space, the linear forms are also precisely the linear functionals.

Definition

It was a body and a vector space, a figure hot now if and linear form if and only if for all vectors and scalars:

The set of all linear forms on a given vector space is the dual space, and thus itself back in a natural way a vector space.

Related terms

Applies specifically and changes to the second condition in, trapping the complex conjugate of designated, one obtains a semi- linear form.

An illustration is linear or semilinear in more than one argument is a sesquilinear form, a bilinear form, or in general a multi- linear form.

Linear form as a tensor

A linear form is a covariant tensor of first stage; they are therefore sometimes also called 1-form. 1-forms are the basis for the introduction of differential forms.

Properties

General properties for linear forms are for example:

• Like any linear map they are determined by their values ​​for any base from complete.
• They are either trivial ( the same everywhere) or surjective.
• Have two of the same from you cores, they differ only by multiplication by a scalar.

Especially for linear functionals also applies:

• They are is continuous if its core is complete.
• Your absolute value is always a semi-norm.
• Linear functionals are exactly the pictures, with a vector and the standard scalar call.