# Vector-valued function

In mathematics, a **vector-valued function** is a function whose target set is a multidimensional vector space . Vector-valued functions are examined in particular in multi-dimensional analysis , differential geometry and functional analysis.

## definition

One function

is called *vector-valued* if its target set is a vector space . In particular, the structure of the definition set is not relevant, only that of the target set.

In many cases, the is used as the vector space ; such functions are then also called *real-vector-valued* . If the vector space is , then the functions are called analog *complex-vector valued* .

## Examples

- The figure defined by

- is a real vector valued function.

- The parametric representation of a curve in two or more dimensions is a real vector-valued function from to .

- A vector-valued function is also called a vector field in this case .

## literature

Otto Forster: Analysis 2 . Differential calculus im , ordinary differential equations. 10th, improved edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-02356-0 , doi : 10.1007 / 978-3-658-02357-7 .