nLab
functional

Idea

The term functional is used in two meanings:

• either a $k$-valued, usually linear, function on a $k$-vector space (or module), where $k$ is a field (or any rig);

  discussed below in In linear algebra and functional analysis

• or a suitably well behave function on some mapping space, for instance a path space or the like

  discussed below in Nonlinear functionals.

Of course the two meanings may overlap.

Physicists and engineers usually speak of functions if the argument is a variable of type the real numbers or complex numbers; but speak of a functional if the type is a mapping space.

Some special cases include

• in linear functional analysis one most often studies continuous linear functionals: linear functions on topological vector spaces. This includes for instance distributions.

• in variational calculus one studies functions on spaces of sections of jet bundles. For example, they appear in physics as action functionals.

• various discretised versions are interesting in finite geometries as well as numerical analysis

In linear algebra and functional analysis

Definition

A functional is a function $V\to k$ from a vector space to the ground field $k$. A linear functional is a linear such function, that is a morphism $V\to k$ in $k$-Vect. In the case that $V$ is a topological vector space, a continuous linear functional is a continuous such map (and so a morphism in the category TVS). When $V$ is a Banach space, we speak of bounded linear functionals, which are the same as the continuous ones.

Remarks

In a sense, linear functionals are co-probes for vector spaces. If the vector space $V$ 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 $0\in k$.

Nonlinear functionals

See

• variational calculus

• action functional

• covariant phase space, path integral