Difference between revisions of "Non-linear functional"
From Encyclopedia of Mathematics
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| − | A special case of a [[Non-linear operator|non-linear operator]] defined on a real (or complex) vector space | + | {{TEX|done}} |
| + | A special case of a [[Non-linear operator|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 | or convex functionals, defined by the condition | ||
| − | + | $$f(\lambda y+(1-\lambda)x)\leq\lambda f(y)+(1-\lambda)f(x),$$ | |
| − | where | + | where $x,y\in X$, $0\leq\lambda\leq1$, and, say, $f(x)=\|x\|$ — the norm of an element in a normed space. |
Latest revision as of 14:55, 14 September 2014
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
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