Problem 1: (a) Let (X, d) be a metric space. Define a function d': X * X → R by setting

d'(x, y) = d(x, y)/1 + d(x, y)

Prove that d' is also a metric on X.

- Hint: Use the fact that d is a metric on X. For the triangle inequality, work backward: start with the desired inequality, express the more complicated side as a single fraction, and then cross-multiply and use other algebra to rewrite the inequality. You should end up with something of the form d(x, z) ≤ d(x, z) + d(z, y) + (something else in term of d), which will be an inequality that you can directly verify using assumptions on d.

(b) Now, let (V,¦¦ . ¦¦) be a formed vector space over R. Suppose that V consists of more than just its zero vector. Prove that for any real number M > 0, there exist points x, y, ∈ V such that ¦¦x - y ¦¦ > M. (Hint: Try taking y to be the zero vector. For x, start with some non-zero vector and rescale appropriately.)

(c) Let (V, ¦¦.¦¦) be as in (b). Recall from class that there is a natural metric d on V given by d(x, y) = ¦¦x-y¦¦. By part (a), there is another metric d' on V given by

d' (x, y) = ¦¦x-y¦¦/1+¦¦x-y¦¦.

Use the result of (b) to show that there does not exist any norm¦¦.¦¦' on X such that d' (x,y) = ¦¦x-y¦¦'. (Hint: Can you bound d'(x, y) from above by a constant?)

Problem 2: Let (f_n) be a sequence of function in the space Cⁿ(a, b), which we take to be a metric space with the metric defined in class. Show that (f_n) converges to a function f ∈ Cⁿ(a, b) with respect to this metric if and only if the sequence of k^th-order derivatives (f_n^(k)) converges uniformly to f^(k) on (a, b) for every k = 0, 1, ... , n.

Problem 3: Let (X, ¦¦.¦¦) be normed vector space over R. As usual, we think of this as a metric space with metric d given by d(x,  y) = ¦¦x - y¦¦.

(a) Suppose that (x_n) and (y_n) are sequences in X such that x_n → x and y_n → y for some vectors x, y ∈ X. Prove that the sequence (x_n + y_n) convergences to x + y.

(b) Suppose that (x_n) is a sequence in X which converges to the vector x, and let c be a fixed real number. Prove that the sequence (cx_n) converges to cx.

Problem 4: Consider the spaces C₂(a, b) and C^o(a, b), equipped with the respective metrics d²_{(a, b)} and d_{(a, b)} defined in class. By their definitions, given a function f ∈ C²(a, b), the second derivative f" is defined and continuous on (a, b), and is therefore an element of C⁰(a, b). Prove that the differential operator D : C²(a, b)  → C⁰(a, b) defined by D(f) = f" is a continuous function.