# Non-linear functional

From Encyclopedia of Mathematics

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.

**How to Cite This Entry:**

Non-linear functional.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Non-linear_functional&oldid=33290

This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article