Lagrange Multipliers and Eigenvalue Problems

In this chapter we shall show what nondegeneracy condition is necessary to justify the Lagrange multiplier rule in the narrower sense for smooth side conditions. Moreover, we will interpret this condition geometrically and explain the connection with manifolds in B-spaces. In this connection, a generalization of the implicit function theorem is the focal point (Theorem 43.C). The central concepts are: (α)Tangent vector, tangent space, and submersion.(β)Regular point of a set.(γ)Manifold.(δ)Tangential mapping.(ε)Critical point of a functional.