 Functional gradient

For a differentiable real-valued functional $F$ on a real Hilbert space $\mathcal{H}$, its functional gradient satisfies $DF(g)[h]=⟨{\nabla }_{\mathcal{H}}F(g),h{⟩}_{\mathcal{H}}$ for every $h\in \mathcal{H}$. A bounded linear derivative has a unique such representative by the Riesz representation theorem. The inner product matters: a formal variational derivative need not equal the functional gradient in another Hilbert space. For $F(g)=\frac{1}{2}{\sum }_i(y_i-g(x_i){)}^2$ in a Reproducing kernel Hilbert space, the reproducing property gives ${\nabla }_{\mathcal{H}}F(g)={\sum }_i(g(x_i)-y_i)k(x_i,\cdot )$. Substitution in the defining inner product proves this formula directly.