APPM 5440: Definitions/Theorems to Know for Exam II 1. Let A ⊆ X. The closure of A, denoted A, is the intersection of all closed sets containing A. (Note: You can use alternate definitions for this or anything on this sheet.) 2. Let X be a metric space and let A ⊆ X. A is dense in X if A = X. 3. Let (X; dX ) and (Y; dY ) be metric spaces. A function f : X ! Y is an isometry if dX (x; y) = dY (f(x); f(y)) for all x; y 2 X. ∗ ∗ 4. A completion of a metric space (X; d) is a pair consisting of a complete metric space (X ; d ) ∗ ∗ and an isometry ' : X ! X such that '(X) is dense in X . 5. A collection of open sets, fGαgα, is an open cover of a set A if A ⊆ [αGα: 6. A subset A of a metric space X is compact if every open cover of A has a finite subcover. 7. Continuous function spaces, inclusions, and properties: •C(X) = set of all real-valued continuous functions on X •Cb(X) = set of all bounded, real-valued, continuous functions on X •C00(X) = set of all real-valued continuous functions on X with compact support (Alternatively, CC (X).) •C0(X) = closure of set of all real-valued continuous functions on X with compact support - Inclusions: C00(X) ⊆ C0(X) ⊆ Cb(X) ⊆ C(X) - Banach Spaces: C0(X) and Cb(X) - If K is compact, C(K) is Banach and C00(K) = C0(K) = Cb(K) = C(K) 8. A sequence of bounded real-valued functions (fn) on a metric space X converges uniformly to a function f if limn!1 jjfn − fjj = 0. (Here, jj · jj is the \uniform norm" or \sup norm".) n 9. The Heine-Borel Theorem: A subset of R is compact if and only if it is closed and bounded. 10. The Arzel´a-AscoliTheorem: A subset of C([a; b]) is compact if and only if it is closed, bounded, and equicontinuous. 11. A family of functions F from a metric space (X; dX ) to a metric space (Y; dY ) is equicon- tinuous if for any " > 0 and any y 2 X, there exists a δ = δ(y; ") > 0 such that dX (x; y) < δ ) dY (f(x); f(y)) < " for all f 2 F. 12. A family of functions F from a metric space (X; dX ) to a metric space (Y; dY ) is uniformly equicontinuous if for any " > 0 there exists a δ = δ(") > 0 such that dX (x; y) < δ ) dY (f(x); f(y)) < " for all f 2 F. 13. A function f : X ! Y is said to be Lipschitz continuous if there exists a 0 < C < 1 such that dY (f(x); f(y)) ≤ C · dX (x; y) for all x; y 2 X. 14. For a Lipschitz continuous function f, the Lipschitz constant is the smallest constant for which the Lipschitz condition holds. It is denoted and defined by d (f(x); f(y)) Lip(f) = sup Y : x6=y fX (x; y) 15. Let (X; dX ) and (Y; dY ) be metric spaces. f : X ! Y is H¨oldercontinuous with exponent α ≥ 0 if α dY (f(x); f(y)) ≤ C[dX (x; y)] for some 0 ≤ C < 1. 16. A subset A of a metric space is precompact if A is compact. 17. Let (X; d) be a metric space. T : X ! X is a contraction mapping if there exists a constant 0 ≤ c < 1 such that d(T (x);T (y)) ≤ c d(x; y) for all x; y 2 X. 18. The Contraction Mapping Theorem: Let (X; d) be a complete metric space. If T : X ! X is a contraction mapping, then T has exactly one fixed point. Furthermore, for any x0 2 X, the sequence x0;T (x0);T (T (x0));::: converges to the fixed point. 19. Theorem: Suppose that f(t; u) is continuous on R. Then for every (t0; u0), there is an open interval I ⊆ R that contains t0 and a continuously differentiable function u : I ! R that satisfies the initial value problem u_(t) = f(t; u(t)); u(t0) = u0: 20. Theorem: Suppose that f(t; u) is continuous in the rectangle R = [t0−a; to+a]×[u0−b; u0+b] and that jf(t; u)j ≤ M for all (t; u) 2 R. Let δ = min(a; b=M). If u(t) is any solution to the initial value problem u_(t) = f(t; u(t)); u(t0) = u0; then ju(t) − u0j ≤ b when jt − t0j ≤ δ. Furthermore, if f is Lipschitz continuous over R in u, uniformly in t (i.e. jf(t; u)−f(t; v)j ≤ C 8 (t; u); (t; v) 2 R) then the solution of the IVP is unique in R. 21. Suppose that k :[a; b] × [a; b] ! R and g :[a; b] ! R are given. The integral equation Z b f(x) = g(x) + k(x; y) f(y) dy a is a Fredholm integral equation. 22. If k :[a; b] × [a; b] ! R and g :[a; b] ! R are continuous and if (Z b ) sup jk(x; y)j dy < 1; a≤x≤b a then there is a unique continuous f :[a; b] ! R that satisfies the Fredholm integral equation. 23. Theorem: Let f : [0; 1] ! R be continuous. The unique solution of the boundary value problem v00(x) = −f(x); v(0) = v(1) = 0 R 1 is given by v(x) = 0 g(x; y) f(y) dy where ( x(1 − y) ; 0 ≤ x ≤ y ≤ 1 g(x; y) = y(1 − x) ; 0 ≤ y ≤ x ≤ 1: 24. Let X be a non-empty set. A topology on X is a collection T of subsets of X with the following properties. (i) ;; X 2 T (ii) arbitrary unions of sets in T are in T (iii) finite intersections of sets in T are in T 25. Let (X; T ) be a topological space. For any x 2 X, V is a neighborhood of x if there exists a G 2 T such that x 2 G ⊆ V . 26. Let (X; TX ) and (Y; TY ) be topological spaces. A function f : X ! Y is continuous at x 2 X if, for every neighborhood W of f(x), there exists a neighborhood V of x such that f(V ) ⊆ W . 27. Theorem: Let (X; TX ) and (Y; TY ) be topological spaces. A function f : X ! Y is continuous −1 if and only if f (G) 2 TX for all G 2 TY . 28. f : X ! Y is a homeomorphism if it is • one-to-one (injective) • onto (surjective) • continuous • f −1 is continuous 29. Let T1 and T2 be two topologies on X. If T1 ⊆ T2, we say that T2 is finer (stronger) than T1 and that T1 is coarser (weaker) than T2. 30. A space is Hausdorff if every pair of points x 6= y has a pair of non-intersecting neighbor- hoods. 31. A space is connected if it can not be decomposed into the union of two disjoint non-empty open sets. p P1 p 32. For 1 ≤ p < 1, ` denotes the space of real valued sequences (xn) such that n=1 jxnj < 1. p P1 p 1=p The ` norm is jjxjjp := ( n=1 jxnj ) . 1 33. ` denotes the space of real valued sequences (xn) such that supn jxnj < 1. 1 The ` norm is jjxjj1 := supn jxnj..