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ANALISI FUNZIONALE FOGLIO DI ESERCIZI N. 2 A.A. 2016/17 Exercise 1. Let (X, d) be a metric space. Define C(X) = {f : X → R : f is continuous} Cb (X) = {f ∈ C(X) : f is continuous and bounded} Cc (X) = {f ∈ C(X) : f is continuous and zero outside a compact subset of X}. For all f ∈ C(X), define the uniform norm of f as . kf k∞ = sup |f (x)|. x∈X (a) Prove that there are metric spaces X for which (C(X), k · k∞ ) is not a metric spaces. (b) Prove that (Cb (X), k · k) is a Banach space. (c) Prove that there are metric spaces X for which Cc (X) is not closed in Cb (X). (d) Define C0 (X) = Cc (X). Find an example of a metric space X such that C0 (X) \ Cc (X) 6= ∅. (e) If X is compact, prove that (C(X), k · k∞ ) is a Banach space and that C(X) = Cb (X) = C0 (X) = Cc (X). Exercise 2. For x, y ∈ R let d(x, y) = |x − y| . 1 + |x − y| Prove that d is a distance on R. Exercise 3. Let a, b ∈ R, a < b. Let α : [a, b] → R be a continuous function with α(x) ≥ β > 0 for some β > 0. For two functions f, g ∈ Cb ([a, b]), define dα (f, g) = sup [α(x)|f (x) − g(x)|] . x∈[a,b] Prove that dα is a metric. Prove that (Cb ([a, b]), dα ) is a complete metric space. Is such distance induced by a norm? Which one? Is the eventual normed space obtained a Banach space? Motivate your answers. 1 2 A.A. 2016/17 Exercise 4. Let Ω ⊂ Rd be an open set. Let {Ωm }m∈N S be an increasing sequence of bounded open sets such that Ωm ⊂ Ωm+1 , Ω = +∞ m=1 Ωm . Prove that the space C(Ω) of continuous functions f : Ω → R is a complete metric space with the metric ρ(f, g) = +∞ X 1 maxx∈Ω |f (x) − g(x)| . 2m 1 + maxx∈Ω |f (x) − g(x)| m=1 Prove also that ρ(fm , f ) → 0 if and only if fm converges to f uniformly on compact subsets of Ω. Exercise 5. Let 1 ≤ p < q ≤ +∞. Prove that `p ⊂ `q (Hint: use that P is +∞ k=1 |yk | < +∞ then yk → 0 as k → +∞). Moreover, prove that if a sequence {xn }n∈N converges to zero in the k · k`p norm then it converges to zero in the k · k`q norm. Exercise 6. Let c = {x ∈ `∞ : limk→+∞ xk exists}. Prove that c is closed in `∞ . Let c0 = {x ∈ `∞ : limk→+∞ xk = 0} equipped with the `∞ -norm. Prove that c0 is closed in c. Exercise 7. Consider the sequence n , for n, k ∈ N. xn,k = 2 k + n2 For all n ∈ N, let xn := {xn,k }k∈N . Prove that xn ∈ `1 ∩ `∞ for all n ≥ 1. Prove that xn → 0 in `∞ but xn does not converge to zero in `1 . Exercise 8. Let (A, d) be a complete metric space, and let Lip(A; R) be the space of Lipschitz continuous functions from A into R. For every f ∈ Lip(A; R), consider the number |f (x) − f (y)| , d(x, y) x,y∈A [f ]Lip := sup and the quantity kf kLip = kf k∞ + [f ]Lip , where kf k∞ is the usual uniform norm of f , i. e. kf k∞ = supx∈A |f (x)|. Prove that k · kLip is a norm on Lip(A; R). Prove that (Lip(A; R), k · kLip ) is a Banach space. Exercise 9. Let A = [a, b] ⊂ R. Let C(A) the (Banach) space of continuous functions on A equipped with the usual uniform k · k∞ norm. Prove with an example that the linear subspace Lip(A) ⊂ C(A) of all Lipschitz continuous functions on A is not closed in (C(A), k · k∞ ). Explain why this is not a contradiction with the fact that the space of Lipschitz functions is a Banach space (and in particular a complete space) from the previous exercise. ANALISI FUNZIONALE FOGLIO DI ESERCIZI N. 2 3 Exercise 10. With the notation of the previous exercise, prove that Lip(A) is dense in (C(A), k · k∞ ) (Hint: for a given continuous function f ∈ C(A), construct an approximation of f in the uniform norm which is a Lipschitz function). Exercise 11. Let A = [a, b] ⊂ R, and let α ∈ (0, 1). Let C 0,α (A) be the space of α-Hölder continuous functions on A, i. e. the set of functions f : A → R such that |f (x) − f (y)| ≤ C|x − y|α for all x, y ∈ A. For all f ∈ C 0,α (A), set |f (x) − f (y)| , |x − y|α x,y∈A kf kC 0,α = kf k∞ + [f ]α . [f ]α := sup Prove that k · kα is a norm on C 0,α . Moreover, prove that the space C 0,α equipped with k · kC 0,α is a Banach space. Exercise 12. Let A = [a, b] ⊂ R, and let α ∈ (0, 1). Let C(A) be the space of continuous functions on A equipped with the uniform norm. Let MK := {f ∈ C(A) : kf kC 0,α ≤ K}. Prove that MK is a compact subset of C(A). Exercise 13. Let (l2 , d) be the complete metric space given by ( ) ∞ X l2 = x = {xn }n∈N : xn ∈ R and x2n < ∞ n=1 and d(x, y) = ∞ X !1 2 |xn − yn |2 . n=1 Show that (1) The set Q = {x ∈ l2 : |xn | ≤ 1 for any n ∈ N} is not bounded. (2) The set B = {x ∈ l2 : ∞ X x2n ≤ 1} n=1 is bounded but not totally bounded. (3) The set H = {x ∈ l2 : |xn | ≤ is compact. 1 for any n ∈ N} n 4 A.A. 2016/17 Exercise 14. Let {fn }n∈N ⊂ C([0, 1]) such that supx∈[0,1] |fn (x)| ≤ 1 for any n ∈ N. Define Fn : [0, 1] → R by Z x fn (t) dt. Fn (x) = 0 Show that the sequence {Fn }n∈N has a subsequence that converges uniformly on [0, 1]. Exercise 15. Let G the following subset of C([0, π]) Z π sin(xy)f (y) dy, f ∈ C([0, π]), kf k∞ ≤ 1 . G = g ∈ C([0, π]; g(x) = 0 Prove the following statement. (1) G is relatively compact in in C([0, π]). (2) G is relatively compact in L1 ([0, π]). Exercise 16. Let f : R3 → R be a continuous function and let A : C([0, 1]) → C([0, 1]) the map defined as follows Z 1 (Au)(x) = f (x, s, u(s)) ds, u ∈ C([0, 1]). 0 Prove the following statement. (1) A is well defined. (2) A is a continuous map. (3) If U is a bounded subset of C([0, 1]), A(U) is relatively compact in C([0, 1]). Exercise 17. Show that C([0, 1]), the space of continuos function from [0, 1] R1 to R, with the norm kf k1 = 0 |f (t)| dt is not complete. Exercise 18. Let X and Y be metric spaces and f : X → Y be a continuous function. Show that if K ⊂ X is compact then f (K) ⊂ Y is compact as well. Exercise 19. Prove that the set B = {f ∈ C([0, 1]) : kf k∞ ≤ 1} is not relatively compact with respect to the sup-norm. Hint: find a sequence in B which admits no convergent subsequences. Exercise 20. Let f : R×R → R be a continuous function in a neighborhood of the point (t0 , y0 ) ∈ R2 . Then, there exists ε > 0 such that the ordinary differential equation ( y 0 (t) = f (t, y(t)) y(t0 ) = x0 has a solution defined on the interval [t0 − ε, t0 + ε].