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