Homework 3, Advanced Calculus 1

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Homework 3, Advanced Calculus 1

For the first two problems, we consider the space for a real number p ≥ 1:

H^p := {f : R → R continuous | Z

R

|f |^pdx < ∞}.

1. Prove that H^pis closed under usual addition and scalar multiplication of functions. That is, for every f, g ∈ H^p and a ∈ R, we have af, f + g ∈ H^p.

2. On H^p, we define d : H^p× H^p → R by

d(f, g) :=

Z

R

|f − g|^p

¹_p . Prove that d satisfies the triangle inequality

d(f, g) ≤ d(f, h) + d(h, g)

and therefore is a metric on H^p. (The other requirements are quite easy to check.) (Hint: For this problem, it is helpful to use the H¨older inequality:

For all p, q ∈ [1, ∞] so that ¹_p + ¹_q = 1, we have Z

R

|f g|dx ≤

Z

R

|f |^p

_p¹ Z

R

|g|^q

¹_q .

Start by usual triangle inequality |F + G|^p ≤ (|F | + |G|)|F + G|^p−1. 3. Prove that for any metric space (X, d), finite sets are closed.

4. Prove that Rⁿ with discrete metric, ie. the one given in Exercise 10 in Rudin, is a bounded set.

Two metrics d₁ and d₂ on a set X are said to be equivalent if there exist constants C₁, C₂ > 0 so that

C₁d₂(x, y) ≤ d₁(x, y) ≤ C₂d₂(x, y) for all x, y ∈ X.

It is straightforward to show that the relation described above is an equivalence relation.

(Problem 9)

5. If two metrics d₁ and d₂ are equivalent on X, prove that they induce the same topology.

That is, a subset E ⊂ X is open with respect to d₁ if and only if E is open with respect to d₂. Prove the same statement with ”open” replaced by ”closed”

6. Prove that on R, the discrete metric (the one in Exercise 10 of Rudin) is not equivalent to the Euclidean metric dEC(x, y) = |x − y|.

7. Prove that on Rⁿ, d_EC is equivalent to the infinity metric:

d∞(x, y) := max_1≤j≤n{|x_j− y_j|}.

8. Prove that the metric d₅ in Exercise 11 of Rudin is not equivalent to d∞. 9. Show that two metrics being equivalent is an equivalence relation.

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