Basic Functional Analysis for the Optimization of Partial Differential Equations

Infinite-dimensional optimization requires – among other things – many results from functional analysis. In this script basics from functional analytic theory is reviewed. The purpose of this work is to give a summary of important facts needed to work in our research group. 1. Functional Analysis – Results and Definitions If M is a set and M1 ⊂ M , the symbol M \M1 represents the complement of M1 in M , i.e. M \M1 = {x ∈ M : x 6∈ M1}. M will always denote the closure of the set M , which is the smallest closed set containing in M . The interior of the set M , M◦, is the largest open set containing in M . The boundary of M is the set ∂M = M \M◦. The set of ordered pairs {(x, y) : x ∈ M1, y ∈ M2} is called the Cartesian product of the sets M1 and M2 and it is denoted M1 ×M2. Let f : M → M1 be a function (or mapping). f(M) will usually called the range of f and will denoted ran (f). The set {x ∈ M : f(x) = 0} is said to be the kernel of f and is denoted ker (f). A function f will be called injective if for each y ∈ ran (f) there is at most one x ∈M such that f(x) = y; f is called surjective if ran (f) = M1. If f is both injective and surjective, we will say it is bijective. Let f : M → M1 and g : M1 → M2 be two functions. The composite mapping r = g ◦ f is defined by r : M →M2, x 7→ r(x) = g(f(x)). Definition 1.1. A (real) linear space is a set, V , over IR, whose elements satisfy the following properties 1) v + w = w + v for all v, w ∈ V , 2) v + (w + u) = (v + w) + u for all v, w, u ∈ V , 3) There is in V a unique element, denoted by 0 and called the zero element, such that v + 0 = v for each v, 4) To each v in V corresponds a unique element, denoted by −v, such that v + (−v) = 0, 5) α (v + w) = α v + αw for all v, w ∈ V and α ∈ IR, 6) (α+ β) v = αv + β v for all v ∈ V and α, β ∈ IR, 7) α (β v) = (αβ) v for all v ∈ V and α, β ∈ IR, 8) 1 · v = v for all v ∈ V , 9) 0 · v = 0 for all v ∈ V . Date: January 8, 2003. 1991 Mathematics Subject Classification. 35Kxx, 46Axx, 46Bxx, 46Cxx, 46Exx, 49Kxx.