# 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 .