Non-linear functional

A special case of a non-linear operator defined on a real (or complex) vector space $X$ and whose values are real (or complex) numbers. Examples of non-linear functionals are the functionals of the calculus of variations,

$$f(x)=\int\limits_a^bF(t,x(t),x'(t))dt,$$

or convex functionals, defined by the condition

$$f(\lambda y+(1-\lambda)x)\leq\lambda f(y)+(1-\lambda)f(x),$$

where $x,y\in X$, $0\leq\lambda\leq1$, and, say, $f(x)=\|x\|$ — the norm of an element in a normed space.

Comments

See also Non-linear functional analysis.