Non-linear functional

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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,


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.


See also Non-linear functional analysis.

How to Cite This Entry:
Non-linear functional. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article