A linear functional on a real vector space 
is a function 
, which satisfies the following properties.
1. 
,
and
2. 
.
When 
is a complex vector space, then 
is a linear map into the complex
numbers.
Generalized functions are a special case of linear functionals, and have a rich theory surrounding them.
See also
Dual Vector Space, Functional, Generalized Function, Linear Function, Vector Space
This entry contributed by Todd Rowland
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Rowland, Todd. "Linear Functional." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LinearFunctional.html