Let $U$ be open in a real Banach space $X$ , and $Y$ be another real Banach space. Let $f\colon U \to \real$ and $g\colon U \to Y$ 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 $\D g(a)\colon X \to Y$ is surjective. Then there exists a Lagrange multiplier vector $\lambda \in Y^*$ such that$$ \D f(a) = \D g(a)^* \lambda = \lambda \circ \D g(a)\,.$$ (The function $\D g(a)^*\colon Y^* \to X^*$ denotes the pullback or adjoint by $\D g(a)$ on the continuous duals, defined by the second equality.)

If $X$ and $Y$ are finite-dimensional, writing out the above equation in matrix form shows that $\lambda$ really is the usual Lagrange multiplier vector. The condition that $\D g(a)$ is surjective means that $\D g(a)$ must have full rank as a matrix.

Bibliography

1

Eberhard Zeidler. Applied functional analysis: main principles and their applications. Springer-Verlag, 1995.