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Metric and Banach Spaces Appendix B Metric and Banach spaces In this appendix, we recall some basic definitions of functional analysis about metric and Banach spaces that are largely employed to prove existence results for partial differential equations. These definitions will in particular provide an answer to the question of ”how close are two objects ?” The topological concepts and the concepts of the measure theory meet in this matter and there are numerous links between these two fields. Contents B.1 Metric spaces . 85 B.2 Complete metric spaces. 87 B.3 Boundedness and compactness . 89 B.4 Normed and Banach spaces . 91 B.5 Schwartz space, Fr´echet space . 96 B.1 Metric spaces Definition B.1 A metric space is a couple (X, d) where X is a set and d is a metric (or a distance) on X, that is a function d : X X R+ such that × → 1. d(x, y) 0 (non-negativity) ≥ 2. d(x, y) = 0 if and only if x = y (identity) 3. d(x, y) = d(y, x) (symmetry) 4. d(x, z) d(x, y) + d(y, z) (triangle inequality). ≤ Remark B.1 Usually, only three conditions are used to define a distance function. Indeed, the first of these conditions is a property that follows from the other three, since: 2d(x, y) = d(x, y) + d(y, x) d(x, x) = 0 . ≥ Definition B.2 Let (X, d) be a metric space and r a stricly positive scalar value. At any point x in a metric space, we define the open ball (resp. closed ball) of radius r about x as the set B(x, r) = y { ∈ X : d(x, y) < r (resp. B (x, r) = y X , d(x, y) r ). } c { ∈ ≤ } 85 86 Chapter B. Metric and Banach spaces These balls generate a topology on X, making it a topological space . A subset Y of X is called an open if it is a union of open balls, its complement is called a closed. Given a non empty and bounded set A of a metric space (X, d). There exists a smallest scalar value δ such that (x, y) A A , d(x, y) δ , ∀ ∈ × ≤ we denote δ(A) this value and we call it the diameter of A. The notion of distance allows to define the concepts of the limit of a sequence and of a continuous function. Definition B.3 (a) Convergence of sequences: Let (xn)n a sequence of elements of a metric set ∈N (X, d) and l X a point. The sequence (xn)n converges towards l if and only if: ∈ ∈N ε > 0 , n , n n d(x , l) < ε . ∀ ∃ 0 ≥ 0 ⇒ n (b) continuity of functions: Consider a function f between two metrics spaces (X, dX ) and (Y, dY ) and a point x X. Then, f is continuous at the point x if: 0 ∈ 0 ε > 0 , δ > 0 , / d (x, x ) < δ d (f(x), f(x )) < ε . ∀ ∃ X 0 ⇒ Y 0 Proposition B.1 (Composition of continuous functions) Let (X, dX ), (Y, dY ) and (Z, dZ ) be three metric spaces and f : X Y , g : Y Z two functions. Suppose f is continuous at a point x and g at → → 0 the point f(x ). Then the function g f is continuous at x . 0 ◦ 0 Proposition B.2 Let (X, dX ) and (Y, dY ) be two metric spaces, f : X Y a function and (xn)n a → ∈N sequence of elements in X. Suppose (xn)n converges to l and the function f is continuous at l. Then, ∈N the sequence of elements of Y defined by (f(xn))n converges to f(l). ∈N We recall three topological concepts in topology. Definition B.4 Suppose (X, d) is a metric space and A X is an arbitrary subset of X. Then, ⊂ (a) The interior of A, denoted as A◦ is the set of points x X such that there exists an open ball B(x, r) ∈ (r > 0) centered at x which is contained in A. (b) The closure of A, denoted as A¯, is the set of points x X such that any open ball B(x, r) (r > 0) ∈ centered at x contains a point of A (the point may be x itself). (c) A is dense in X if and only if A¯ = X. Remark B.2 (i) the set A is open if and only if A = A◦ . In particular, the interior of the empty set if the empty set. (ii) the set A is closed if and only if A¯ = A. In particular, the closure of the empty set is the empty set and the closure of A itself is A. Proposition B.3 Let A be a subset of a metric space (X, d). A point x belongs to A¯ if and only if there exists a sequence (an)n of elements of A such that lim an = x. ∈N n →∞ B.2. Complete metric spaces. 87 Definition B.5 Let (xn)n be a sequence in a metric space (X, d). We define the set of all closure ∈N values of (xn)n , denoted as Clo(xn) as: ∈N def def Clo(x ) = A¯ with A = x , m n . n n n { m ≥ } n !∈N Then , we have a concept very important in numerical analysis. Theorem B.1 Let (xn)n be a sequence in a metric space (X, d). A point l X belongs to Clo(xn) if ∈N ∈ and only if there exists a strictly increasing function φ : N N ( n N, φ(n) > n) such that: → ∀ ∈ lim xφ(n) = l . n →∞ Definition B.6 Let (X, d ) and (Y, d ) be two metric spaces and let consider a function f : X Y . X Y → Then, (a) the function is uniformly continuous if and only if: ε > 0 , α > 0 , (x, x%) X , d (x, x%) < α d (f(x), f(x%)) < ε , ∀ ∃ ∀ ∈ X ⇒ Y (b) the function f is locally H¨older continuous (satisfies a H¨older condition) of exponent α [0, 1] if ∈ and only if there exists a nonnegative constant k such that: α (x, x%) X X , d (x, x%) 1 d (f(x), f(x%)) k d (x, x%) . ∀ ∈ × X ≤ ⇒ Y ≤ X Remark B.3 (see also Definition (C.1)). In the previous definition, (i) If α = 1, the function f is locally k-Lipschitz continuous. (ii) If α = 1 and if the condition d (x, x ) 1 can be omited, the function f is globally k-Lipschitz X % ≤ continuous. Definition B.7 Given a subset A X of a metric space (X, d) and x X. The distance between the ⊂ ∈ point x and the set A is the infimum of the distances between the point and those in the set: def d(x, A) = inf d(x, a) . a A ∈ + Proposition B.4 Given A (X, d). The function dA : X R , x d(x, A) is k-Lipschitz with the ⊂ → +→ constant k = 1. B.2 Complete metric spaces. Definition B.8 Let (X, d) be a metric space. A Cauchy sequence in X is a sequence (xn)n of elements ∈N of X such that: ε > 0 , n0 N , n n0 , m n0 , d(xn, xm) ε . ∀ ∃ ∈ ∀ ≥ ∀ ≥ ≤ In particular, any convergent sequence is a Cauchy sequence thanks to the triangle inequality. Proposition B.5 Any Cauchy sequence in a metric space is bounded. 88 Chapter B. Metric and Banach spaces Proof. If (xn)n is a Cauchy sequence, there exists n0 N such that for all n n0, and for all m n0, ∈N ∈ ≥ ≥ we have d(xn, xm) 1 + max d(xn, xm). The results follows. 2 ! ≤ (n,m) 0,... n0 ∈{ } Notice that given a Cauchy sequence (xn)n and a continuous function f then (f(xn))n is not ∈N ∈N necessarily a Cauchy sequence. However, we have Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) (Y, dY ). If (xn)n is a Cauchy sequence of X, then f(xn)n is a Cauchy → ∈N ∈N sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about convergent sequences and Cauchy sequences. (a) In a metric space (X, d), any convergente sequence is a Cauchy sequence and conversely, any Cauchy sequence having a closure value l converges to l. (b) In any metrix space (X, d), a Cauchy sequence which has convergent subsequence with limit l is itself convergent with the same limit l. These two properties yield one standard proof of the completness of the real numbers (with the Bolzano- Weierstrass theorem). Definition B.9 A metric space (X, d) in which every Cauchy sequence converges (has a limit in X) is called complete. A Polish space is a separable complete metric space. The advantage of working wih Cauchy sequences and with the completness concept is provided by the Banach spaces. The following theorem will be used to show the existence of a unique solution for ordinary differential equation (see Cauchy-Lipschitz theorem). Theorem B.3 (Banach fixed-point) Given f : (X, d) (X, d) a contraction map, i.e. a function → such that there exists a real number k [0, 1[ such that ∈ d(f(x), f(y)) k d(x, y) . ≤ Then, there exists a unique fixed point z such that f(z) = z. Proof. (Picard’s iterative method) Given x0 X, we consider the sequence (xn)n defined as ∈ ∈N xn+1 = f(xn). We will show that this sequence converges toward the fixed point z, for all x0.
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