Banach Families And The Implicit Function Theorem
Jean‐François Mertens, Anna Rubinchik · RePEc: Research Papers in Economics · 2011
Abstract. We generalise the classical implicit function theorem (IFT) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms. Notation. For Banach spaces X and Y, L(X,Y) is the Banach space of continuous linear maps from X to Y; so L(X,X) is the Banach algebra of operators on X. X × Y and1 X ∩ Y have by default the maximum norm. Definition 1. Fix a pointed set A, i.e. a pair (A,α0) with α0 ∈ A. Fix also a class of subsets A of A with α0 ∈ S ∀S ∈ A. A Banach family (B,A), or shortly B, is a Banach space (B, ‖·‖) endowed with a collection of pseudo-norms (‖·‖α)α∈A, s.t. ‖·‖α0 = ‖·‖, where we allow pseudo-norms to take infinite values.2 Bα = {x ∈ B | ‖x‖α <∞}. 3,4 For 2 Banach families X and Y, and a linear map ϕ from X to Y, let ‖ϕ‖α = sup{‖ϕ(x)‖α | ‖x‖α ≤ 1}, and, for S ∈ A, ‖ϕ‖S = supα∈S‖ϕ‖α; and let LA(X,Y) = {ϕ ∈ L(X,Y) | ∀S ∈ A, ‖ϕ‖S <∞}, endowed with the family of norms (‖·‖S)S∈A. Remark 1. Nothing prevents to endow the same Banach space with 2 different Banach family structures; those should however be distinguished notationally then. E.g., Rn will denote denote this space with the constant family of pseudo-norms, while (Rn, {0}) will be used when, for α 6 = α0, Rnα = {0}. Remark 2. The main intent is to be able to view B also as some sort of Bα-manifold; i.e., to speak of α-neighbourhoods of points in B. Remark 3. As is clear from the definition, the purpose of constructing Banach families is to get operator norms, later used to formulate the ift. One could, con-ceivably, use a classical ift for each of the norms α ∈ A and get as final conclusion: