linear functional

Let V be a vector spaceMathworldPlanetmath over a field K. A linear functionalMathworldPlanetmathPlanetmathPlanetmath (or linear form) on V is a linear mapping ϕ:VK, where K is thought of as a one-dimensional vector space over itself.

The collectionMathworldPlanetmath of all linear functionals on V can be made into a vector space by defining additionPlanetmathPlanetmath and scalar multiplication pointwise; this vector space is called the dual spaceMathworldPlanetmathPlanetmathPlanetmath of V.

The term linear functional derives from the case where V is a space of functions (see the entry on functionalsMathworldPlanetmathPlanetmath ( 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