# linear functional

Let $V$ be a vector space over a field $K$. A linear functional (or linear form) on $V$ is a linear mapping $\phi\colon V\to K$, 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 LinearFunctional 2013-03-22 12:13:54 2013-03-22 12:13:54 yark (2760) yark (2760) 9 yark (2760) Definition msc 15A99 linear form DualSpace CalculusOfVariations AdditiveFunction2 MultiplicativeLinearFunctional