nLab
nonlinear functional

Contents

Idea

Often a (smooth) function on an infinite-dimensional manifold (or on a more general smooth space) is called a functional, especially when scalar-valued. A central example are the action functionals in physics. The calculus of variations is largely about such functionals.

To distinguish from the specifically linear functionals in functional analysis, one might speak here of nonlinear functionals. However, in practice the context is usually understood and the adjective “nonlinear” is rarely used explicitly. There are also nonlinear functionals in functional analysis, such as quadratic forms.

The reason for the name is that (like the infinite-dimensional vector spaces of functional analysis), these infinite-dimensional manifolds were originally spaces of functions (typically between finite-dimensional manifolds), and functions of functions are the original meaning of functional.

Last revised on October 14, 2013 at 12:54:45. See the history of this page for a list of all contributions to it.