For a differentiable real-valued function on an open convex subset U of R the Legendre conjugate of the pair (U, f) is defined to be the pair (V, g), where V is the image of U under the gradient mapping Df, and g is the function on V given by the formula g ( y ) = ⟨ y , x ⟩ − − f ( x ) , x = ( D f ) − − 1 ( y ) where ⟨ u , v ⟩ = ∑ ∑ k = 1 n u k ⋅ ⋅ v k