# linear functional

Let $V$ be a vector space^{} over a field $K$.
A *linear functional ^{}* (or

*linear form*) on $V$ is a linear mapping $\varphi :V\to K$, where $K$ is thought of as a one-dimensional vector space over itself.

The collection^{} of all linear functionals on $V$
can be made into a vector space
by defining addition^{} and scalar multiplication pointwise;
this vector space is called the dual space^{} of $V$.

The term linear functional derives from
the case where $V$ is a space of functions
(see the entry on functionals^{} (http://planetmath.org/Functional)).
Some authors restrict the term to this case.

Title | linear functional |
---|---|

Canonical name | LinearFunctional |

Date of creation | 2013-03-22 12:13:54 |

Last modified on | 2013-03-22 12:13:54 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 15A99 |

Synonym | linear form |

Related topic | DualSpace |

Related topic | CalculusOfVariations |

Related topic | AdditiveFunction2 |

Related topic | MultiplicativeLinearFunctional |