LagrangeMultipliersInBanachSpaces 2005-08-16 21:28:39 2005-08-17 08:34:27 Theorem Lagrange multiplier method \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{amsthm} \usepackage{enumerate} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} % define commands here \newcommand{\complex}{\mathbb{C}} \newcommand{\real}{\mathbb{R}} \newcommand{\rat}{\mathbb{Q}} \newcommand{\nat}{\mathbb{N}} \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{\absW}[1]{\left\lvert#1\right\rvert} \providecommand{\absB}[1]{\Bigl\lvert#1\Bigr\rvert} \providecommand{\norm}[1]{\lVert#1\rVert} \providecommand{\normW}[1]{\left\lVert#1\right\rVert} \providecommand{\normB}[1]{\Bigl\lVert#1\Bigr\rVert} \providecommand{\defnterm}[1]{\emph{#1}} \DeclareMathOperator{\D}{D} \DeclareMathOperator{\linspan}{span} 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 Fr\'echet 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. \begin{thebibliography}{3} \bibitem{Zeidler} Eberhard Zeidler. {\it Applied functional analysis: main principles and their applications}. Springer-Verlag, 1995. \end{thebibliography}