SOME QUESTIONS FOR MATH 766, SPRING Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm

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1 SOME QUESTIONS FOR MATH 766, SPRING 2016 SHUANGLIN SHAO Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm f C = sup f(x). 0 x 1 Prove that C([0, 1]) is a complete normed space. Assuming that the Weierstrass theorem that the set of polynomials is dense in C([0, 1]), prove that C([0, 1]) is separable. Let f n (x) = x n, 0 x 1. Then f n C([0, 1]). But {f n } is not a Cauchy sequence in C([0, 1]). Show that C([0, 1]) is not compact. Question 2. The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence. Let {f n } be a point-wise bounded sequence of complex functions on a countable set E. Use the Bolzano- Weierstrass theorem to show that {f n } has a subsequence {f nk } such that {f nk (x)} converges for every x E. Question 3. The sequence {f n } is said to be equicontinuous if for any ɛ > 0, there exists δ > 0 such that for x y < δ, f n (x) f n (y) < ɛ for all n. Let {f n } be a point-wise bounded and equicontinuous sequence of functions on a compact set K. Suppose also K contains a dense countable set E. Prove that f n contains a subsequence that converges uniformly on K. Question 4. Let K be a compact set. Let f be a continuous function that is 1 1 on K. Prove that the inverse function of f on f(k) is also continuous. Question 5. [0, 1]. Define Let f C 1 ([0, 1]), i.e., f is continuously differentiable on g(x, y) = { f(x) f(y) x y, if x y, f (x), ifx = y. 1

2 Prove that g is continuous on [0, 1] [0, 1]. If f (x 0 ) 0, prove that there exists a neighborhood V of x 0 in [0, 1] such that f is 1 1 on V. Let h : f(v ) V be the inverse function. Prove that h is continuously differentiable. Question 6. Let f n be a sequence of differentiable functions on [a, b] with a < b satisfying that f n (a) = 0 and f n is integrable on [a, b]. Suppose that {f n} is uniformly convergent on [a, b]. Prove that {f n } is also uniformly convergent on [a, b], and for all x [a, b], ( lim n f n) (x) = lim n f n(x). Question 7. Let f C([0, 1]). Define F (z) = 1 0 f(x), z / [0, 1]. x z Prove that F is a real analytic function on R \ [0, 1]. Question 8. [0, 1]. Let f n f uniformly on [0, 1] and each f n is integrable on Prove that f is integrable. Construct an example of a sequence of bounded functions showing the conclusion in the first part can fail if we only assume that f n f point-wise on [0, 1]. Question 9. that Assume that φ is a continuous real function on (a, b) such φ( x + y ) φ(x) φ(y) for all x, y (a, b). Prove that φ is convex, i.e., φ(λx + (1 λ)y) λφ(x) + (1 λ)φ(y) for all λ [0, 1], x, y (a, b). Question 10. Let f,g be two real analytic functions on (a, b). Prove that f ± g, fg are real analytic on (a, b). Question 11. Let f(x) = e 1 x 2 for x 0, and f(0) = 0. Prove that f is infinitely differentiable on R but f is not real analytic. Question 12. An open interval on R is in form of (a, b), where a, b can take or. Prove that every open set on R is countable disjoint union of open intervals. 2

3 Question 13. on R is R. Use # 12 to show that a nonempty closed and open set E Question 14. Let f be a nonzero real analytic function on R. Prove that the zero set of f, N (f) = {x : f(x) = 0} is countable. Question 15. Let (X, ρ) be a metric space. Let E, F X. Define the distance between E and F to be and Prove the following. d(e, F ) = inf{ρ(x, y) : x E, y F }; d(x, E) = inf{ρ(x, y) : y E}. If E is compact in X, x / E, then d(x, E) > 0. If E and F are compact in X and E F =, the d(e, F ) > 0. Let E be a bounded open set in the standard Euclidean space R n. Let K E be compact. Prove that there exists an open set V in E such that K V V E. Question 16. Prove that the standard Euclidean space R n is separable. Question 17. Let A, B be open and bounded sets in the standard Euclidean space R. Let Prove that A + B is open. A + B = {x + y : x A, y B}. Question 18. Suppose that f is a differentiable real function in an open set E R n and that f has a local maximum at a point x E. Prove that f (x) = 0. Question 19. Suppose that f is differentiable mapping of R into R 3 such that f(t) = 1 for every t. Prove that Interpret this result geometrically. f (t) f(t) = 0. Question 20. Define f(0, 0) = 0 and f(x, y) = x3 if (x, y) (0, 0). x 2 +y 2 Prove that D 1 f and D 2 f are bounded functions in R 2. Therefore f is continuous. 3

4 Let u be any unit vector in R 2. Define the directional derivative of f at (x, y) along u is f((x, y) + hu) f(x, y) D u f(x, y) =. h Show that D u f(0, 0) exists, and that its absolute value is at most 1. Let γ be a differentiable mapping of R 1 into R 2 with γ(0) = (0, 0), and γ (0) > 0. Define g(t) = f(γ(t)). Prove that g is differentiable at every t R 1. Prove that f is not differentiable at (0, 0). Question 21. Define f in R 3 by f(x, y 1, y 2 ) = x 2 y 1 + e x + y 2. Show that f(0, 1, 1) = 0, D 1 f(0, 1, 1) 0 and there exists a differentiable function g in some neighborhood of (1, 1) in R 2 such that g(1, 1) = 0 and f(g(y 1, y 2 ), y 1, y 2 ) = 0. Find D 1 g(1, 1) and D 2 g(1, 1). Question 22. It is known that continuous functions map connected sets to connected sets. Use this to prove the intermediate value theorem for a real continuous function f on [ 1, 1]. Question 23. Let x = (x 1,, x n ), and 2 p <. Define ( n ) 1/p x p = x i p ; for p =, x = max 1 i n x i. Prove that for 2 p, i=1 2 2 p x p x 2 x 1 p. Furthermore, what is x 0 such that the equality here holds? Suppose n = 2. Let x = 1 and x 1, x 2 ɛ for some sufficiently small 0 < ɛ < 1. Prove that there exists C = C(p) independent of ɛ and x such that x p 1 Cɛ. Question 24. Let (X, ρ) be a complete metric space, and let Φ : X X be a strict contraction on X, i.e., there exists 0 < c < 1 such that for all u, v X. ρ(φ(u), Φ(v)) cρ(u, v) Prove that there exists a unique x X such that Φ(x) = x. 4

5 Let u = Φ(u). Prove that for any v X, ρ(v, u) 1 ρ(v, Φ(v)). 1 c Question 25. Let f : R 2 R. Suppose that D 1 f, D 2 f and D 12 f exist on R 2. Suppose also that D 12 f is continuous on x 0 R 2. Prove that D 21 f exists at x 0 and D 21 f(x 0 ) = D 12 f(x 0 ). Department of Mathematics, KU, Lawrence, KS address: slshao@ku.edu 5