Metric and Topological Spaces

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## Exercises 3

1. If f, g are continuous functions on an interval [a, b], prove that d1(f, g) (b - a)d(f, g).
Hence prove that if a sequence (fn) in C[a, b] converges to a function f in d it also converges to f in d1.

2. A sequence (xi) in a metric space (X, d) is called a Cauchy sequence if:
Given > 0, there exists N N such that m, n > N d(xm, xn) < .
Let (fn) be the sequence of functions on [-1, 1] defined by
fn(x) = 0 for x 0, fn(x) = nx for 0 < x < 1/n and fn(x) = 1 for x 1/n.
Sketch the graphs of a typical pair of functions fm and fn and hence or otherwise prove that the sequence (fn) is a Cauchy sequence in C[-1, 1] with the metric d1.
Calculate the pointwise limit of the sequence (fn) and hence prove that the sequence (fn) is not a convergent sequence in C[-1, 1] under this metric.

3. Let f be a map from a metric space X to a metric space Y. If the metric on X is the discrete metric, prove that f is continuous. If the metric on Y is discrete, is f necessarily continuous ?

4. This question tries to show you why the metric d2 on a space of continuous functions is an easier one to work with than d1 or d.

Find the best-fit straight-line through the origin to the function x2 in the metric d1 .
That is, find the value of a which minimises d1(x2, ax).

For this you have to minimise the shaded area. Calculate this area by first finding the x-coordinate of the point P in terms of a.

Find the best-fit straight-line through the origin to the function x2 in the metric d .
That is, find the value of a which minimises d(x2, ax).
For this you have to choose a so that the "vertical distance between the graphs" is a minimum. From the picture, you can see that the best you can do is to adjust the value of a so that the two dark lines have equal length.

Calculate d2(x2, ax) and then use the usual calculus method for finding the maximal value of a function to find the best-fit straight-line through the origin to the function x2 in the metric d2 .

SOLUTIONS TO WHOLE SET
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JOC February 2004