The term functional is used in two meanings:
discussed below in In linear algebra and functional analysis
discussed below in Nonlinear functionals.
Of course the two meanings may overlap.
Some special cases include
various discretised versions are interesting in finite geometries as well as numerical analysis
A functional is a function from a vector space to the ground field . A linear functional is a linear such function, that is a morphism in -Vect. In the case that is a topological vector space, a continuous linear functional is a continuous such map (and so a morphism in the category TVS). When is a Banach space, we speak of bounded linear functionals, which are the same as the continuous ones.
In a sense, linear functionals are co-probes for vector spaces. If the vector space in question has finite dimension and is equipped with a basis, then all linear functionals are linear combinations of the coordinate projection?s. These projections are known as the dual basis.
In infinite-dimensional topological vector spaces, the notion of dual basis breaks down once spaces more general than Hilbert spaces are considered. But for locally convex spaces, the Hahn–Banach theorem ensures the existence of ‘enough’ continuous linear functionals. Among non-LCSes, however, there are examples such that the only continuous linear functional is the constant map onto .