Lagrange multipliers on Banach spaces

Let U be open in a real Banach spaceMathworldPlanetmath X, and Y be another real Banach space. Let f:U and g:UY be continuously differentiable functions.

Suppose that a is a minimum or maximum point of f on M={xU:g(x)=0}, and the Fréchet derivative Dg(a):XY is surjective. Then there exists a Lagrange multiplierMathworldPlanetmath vector λY* such that


(The function Dg(a)*:Y*X* denotes the pullback or adjointPlanetmathPlanetmath by Dg(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 λ really is the usual Lagrange multiplier vector. The condition that Dg(a) is surjective means that Dg(a) must have full rank as a matrix.


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