# Lagrange multipliers on Banach spaces

Let $U$ be open in a real Banach space $X$, and $Y$ be another real Banach space. Let $f\colon U\to\mathbb{R}$ 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 $\operatorname{D}g(a)\colon X\to Y$ is surjective. Then there exists a Lagrange multiplier vector $\lambda\in Y^{*}$ such that

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

(The function $\operatorname{D}g(a)^{*}\colon Y^{*}\to X^{*}$ denotes the pullback or adjoint by $\operatorname{D}g(a)$ 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 $\operatorname{D}g(a)$ is surjective means that $\operatorname{D}g(a)$ must have full rank as a matrix.

## References

• 1 Eberhard Zeidler. Applied functional analysis: main principles and their applications. Springer-Verlag, 1995.
Title Lagrange multipliers on Banach spaces LagrangeMultipliersOnBanachSpaces 2013-03-22 15:28:30 2013-03-22 15:28:30 stevecheng (10074) stevecheng (10074) 5 stevecheng (10074) Theorem msc 49-00 msc 49K35