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 $\lambda \in Y^{*}$ such that

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

(The function $\mathrm{D}g{(a)}^{*}:Y^{*}\to X^{*}$ denotes the 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 above equation in matrix form shows that $\lambda$ really is the usual Lagrange multiplier vector. The condition that $\mathrm{D}g(a)$ is surjective means that $\mathrm{D}g(a)$ must have full rank as a matrix.

Title	Lagrange multipliers on Banach spaces
Canonical name	LagrangeMultipliersOnBanachSpaces
Date of creation	2013-03-22 15:28:30
Last modified on	2013-03-22 15:28:30
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	5
Author	stevecheng (10074)
Entry type	Theorem
Classification	msc 49-00
Classification	msc 49K35