Editing Legendre transformation (section)

===Definition in n-dimensional real space===
The generalization to convex functions <math>f:X \to \R</math> on a [[convex set]] <math>X \sub \R^n</math> is straightforward: <math>f^*:X^* \to \R</math> has domain
<math display="block">X^*= \left \{x^* \in \R^n:\sup_{x\in X}(\langle x^*,x\rangle-f(x))<\infty \right \}</math>
and is defined by
<math display="block">f^*(x^*) = \sup_{x\in X}(\langle x^*,x\rangle-f(x)),\quad x^*\in X^* ~,</math>
where <math>\langle x^*,x \rangle</math> denotes the [[dot product]] of <math>x^*</math> and <math>x</math>.

The Legendre transformation is an application of the [[Duality (projective geometry)|duality]] relationship between points and lines. The functional relationship specified by <math>f</math> can be represented equally well as a set of <math>(x,y)</math> points, or as a set of tangent lines specified by their slope and intercept values.