Let X be the subset of  consisting of all sequences x such that  converges. Then the formu...

Let X be the subset of  consisting of all sequences x such that  converges. Then the formula

defines a metric on X. (See Exercise.) On X we have the three topologies it inherits from the box, uniform, and product topologies on . We have also the topology given by the metric d, which we call the . (Read “little ell two.”)

Let X denote the subset of  consisting of all sequences (x1, x2, …) such that ** converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise of the next section.)

Show that if d is a metric for X, then

is a bounded metric that gives the topology of X. [Hint: If f(x) = x/(1 +x) for x > 0, use the mean-value theorem to show that f(a + b) - f (b) ≤ f(a).]

(a) Show that if , then  converges. [Hint: Use (b) of Exercise to show' that the partial sums are bounded.]

Show that the euclidean metric don  is a metric as follows: If  and , define

(a) Show that x • (y + z) = (x • y) + (x • z).

(b) Show that |x•y| ≤ ||x|| ||y||. [Hint: If , let a = 1/||x|| and b = 1/||y||, and use the fact that ||ax ±by|| ≥ 0.1

(c) Show that ||x + y|| ≤ ||x|| + ||y||. [Hint: Compute (x + y)  (x + y) and apply (b).]

(d) Verify that d is a metric.

(b) Let . Show that if , then so are x + y and cx.

(c) Show that

is a well-defined metric on X.

(a) Show that on X, we have the inclusions

(b) The set  of all sequences that are eventually zero is contained in X. Show that the four topologies that  inherits as a subspace of X are all distinct.

(c) The set

is contained in X; it is called the Hilbert cube. Compare the four topologies that H inherits as a subspace of X.