Functional gradient
= Functional gradient
{title2=$\nabla_{\mathcal H}F(g)$}
For a differentiable real-valued functional $F$ on a real <Hilbert space> $\mathcal H$, its functional gradient satisfies $DF(g)[h]=\langle\nabla_{\mathcal H}F(g),h\rangle_{\mathcal H}$ 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 $F(g)=\frac12\sum_i(y_i-g(x_i))^2$ 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.