APPM 5440: Definitions/Theorems to Know for Exam II

1. Let A ⊆ X. The 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 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 ∈ X.

∗ ∗ 4. A completion of a metric space (X, d) is a pair consisting of a (X , d ) ∗ ∗ and an isometry ϕ : X → X such that ϕ(X) is dense in X .

5. A collection of open sets, {Gα}α, is an open of a set A if

A ⊆ ∪αGα.

6. A A of a metric space X is compact if every open cover of A has a finite subcover.

7. 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 (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 of bounded real-valued functions (fn) on a metric space X converges uniformly to a function f if limn→∞ ||fn − f|| = 0. (Here, || · || 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 ε > 0 and any y ∈ X, there exists a δ = δ(y, ε) > 0 such that

dX (x, y) < δ ⇒ dY (f(x), f(y)) < ε

for all f ∈ 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 ε > 0 there exists a δ = δ(ε) > 0 such that

dX (x, y) < δ ⇒ dY (f(x), f(y)) < ε

for all f ∈ F.

13. A function f : X → Y is said to be Lipschitz continuous if there exists a 0 < C < ∞ such that dY (f(x), f(y)) ≤ C · dX (x, y) for all x, y ∈ 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 < ∞.

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 ∈ 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 ∈ 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 |f(t, u)| ≤ M for all (t, u) ∈ 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 |u(t) − u0| ≤ b when |t − t0| ≤ δ. Furthermore, if f is Lipschitz continuous over R in u, uniformly in t (i.e. |f(t, u)−f(t, v)| ≤ C ∀ (t, u), (t, v) ∈ 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 |k(x, y)| 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 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-. A topology on X is a collection T of of X with the following properties.

(i) ∅, X ∈ 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 . For any x ∈ X, V is a neighborhood of x if there exists a G ∈ T such that x ∈ G ⊆ V .

26. Let (X, TX ) and (Y, TY ) be topological spaces. A function f : X → Y is continuous at x ∈ 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) ∈ TX for all G ∈ 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 of two disjoint non-empty open sets.

p P∞ p 32. For 1 ≤ p < ∞, ` denotes the space of real valued sequences (xn) such that n=1 |xn| < ∞. p P∞ p 1/p The ` norm is ||x||p := ( n=1 |xn| ) .

∞ 33. ` denotes the space of real valued (xn) such that supn |xn| < ∞. ∞ The ` norm is ||x||∞ := supn |xn|.