View source for Non-linear functional

A special case of a [[Non-linear operator|non-linear operator]] defined on a real (or complex) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671201.png" /> and whose values are real (or complex) numbers. Examples of non-linear functionals are the functionals of the calculus of variations,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671202.png" /></td> </tr></table>

or convex functionals, defined by the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671203.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671205.png" />, and, say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067120/n0671206.png" /> — the norm of an element in a normed space.



====Comments====
See also [[Non-linear functional analysis|Non-linear functional analysis]].