Math 447: Real Variables Homework 6

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Homework 6

Math 447: Real Variables

Exercise 1 Let (X, dX) and (Y, dY) be two metric spaces. Suppose there are positive constants C1 and C2 and a bijective function f: X → Y such that

C1 dX(x1, x2) ≤ dY (f(x1), f(x2)) ≤ C2 dX(x1, x2)

for all x1, x2 ∈ X.

Prove that if X is complete then Y is complete.

Exercise 2 [Ross, Ex. 24.2] For x ∈ [0, ∞), let f(x) = n/x .

(a) Find (with proof) f(x) = lim fn(x).

(b) Determine (with proof) whether fn → f uniformly on [0, 1].

(c) Determine (with proof) whether fn → f uniformly on [0, ∞).

Exercise 3 For any metric space X, recall that if A ⊂ B ⊂ X then A is dense in B if for all open subsets O ⊂ X if O ∩ B = ∅ then O ∩ A = ∅.

(a) Let X and Y be metric spaces, f : X → Y continuous and surjective, and A ⊂ B ⊂ X. Prove that if A is dense in B then f(A) is dense in f(B).

(b) Assume that f is also injective (hence bijective). Is the converse true: if f(A) is dense in f(B) is A necessarily dense in B? Prove or find a counterexample.

Exercise 4 Let D ⊂ R be a nonempty subset that is also closed under addition and inverses: (∀x, y ∈ D) x+y ∈ D and −x ∈ D (that is, D is an “abelian subgroup” of R).

Define

√ D = { x ∈ R | x2 ∈ D }.

(a) Prove that √ D has the same cardinality as D (recall that two sets have the same cardinality if there is a bijection between them).

(b) Prove that D is dense in R if and only if √ D is dense in R.

(c) Let S 1 (r) be the circle in R 2 with center (0, 0) and radius r ∈ R+:

S1 (r) = { (x, y) ∈ R2 | x2 + y2 = r2 }.

Prove that S 1 (r) ∩ ( √ D × √ D) is empty if r /∈ √ D.

(d) Assume D is dense and r ∈ √ D. Prove that S1 (r) ∩ ( √ D × √ D) is dense in S1 (r).

Exercise 5 Let S n(r) be a sphere of radius r in R n+1:

Sn (r) = { (x1, . . . , xn+1) ∈ Rn+1 | x21 + · · · + x2n+1 = r2 }.

Prove that S n(r)∩(√ D
n+1 is dense if r ∈ √ D and empty if r /∈ √ D (where

Hint: Use induction and Ex. 4.