View source for Potential operator

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741301.png" /> of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741302.png" /> into the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741303.png" /> that is the gradient of some functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741304.png" />, i.e. is such that

<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/p/p074/p074130/p0741305.png" /></td> </tr></table>

For instance, any bounded [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741306.png" /> defined on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741307.png" /> is potential:

<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/p/p074/p074130/p0741308.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Vainberg,   "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Gajewski,   K. Gröger,   K. Zacharias,   "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag  (1974)</TD></TR></table>