Lagrange multipliers on Banach spaces

Let $U$ be open in a real Banach space $X$,and $Y$ be another real Banach space.Let $f:U\to ℝ$ and $g:U\to Y$be continuously differentiable functions.

Suppose that $a$ is a minimum or maximum point of $f$on $M=\{x\in U:g(x)=0\}$,and the Fréchet derivative $\mathrm{D}g(a):X\to Y$is surjective. Then there exists a Lagrange multiplier vector$\lambda \in Y^{*}$suchthat

$$\mathrm{D}f(a)=\mathrm{D}g{(a)}^{*}\lambda =\lambda \circ \mathrm{D}g(a).$$

(The function $\mathrm{D}g{(a)}^{*}:Y^{*}\to X^{*}$ denotesthe pullback or adjoint by $\mathrm{D}g(a)$ on the continuous duals,defined by the second equality.)

If $X$ and $Y$ are finite-dimensional, writing out the aboveequation in matrix form shows that $\lambda$ reallyis the usual Lagrange multiplier vector. The conditionthat $\mathrm{D}g(a)$ is surjective means that $\mathrm{D}g(a)$must have full rank as a matrix.