SOLVED EE6415 Tutorial 4

1. Let X be the set of all ordered triples of zeros and ones. Show that X consists of eight elements and a metric d on X is defined by d(x, y) = number of places where x and y have different entries. For example, d(010, 111) = 2. (This metric is called Hamming distance). 2. Show that the function d on the set X defined by d(x, y) = Z b a |x(t) − y(t)|dt is a metric, where X is the set of all real-valued functions x, y, · · · which are functions of an independent real variable t and are defined and continuous on a given closed interval J = [a, b]. 3. Consider the space of all sequences x = (ζi), y = (ηi). Prove that d(x, y) = X∞ j=1 1 2 j |ζj − ηj | 1 + |ζj − ηj | is a metric. Further, show that d2(x, y) = X∞ j=1 rj |ζj − ηj | 1 + |ζj − ηj | is a metric for any sequence (rj ) for which every element is positive, and Prj converges. 4. Show that d(x, y) = p |x − y| is a metric on the set of Real Numbers. 5. Show that a Cauchy sequence is bounded. Is boundedness of a sequence in a metric space sufficient for the sequence to be Cauchy? Convergent? 6. Let d be a metric on X. Determine all constants k such that (a) kd 1 (b) d + k is a metric on X 7. Does d(x, y) = (x − y) 2 define a metric on the set of all real numbers? 8. The triangle inequality has several useful consequences. Show that the following inequalities are true for any metric d (a) |d(x, y) − d(z, w)| ≤ d(x, z) + d(y, w) (b) |d(x, z) − d(y, z)| ≤ d(x, y) 9. Consider the normed linear vector space of rational numbers Q with norm ∥x∥ = |x|. For each of the sequences an given next, find whether an is (a) convergent in Q (b) a Cauchy sequence. (a) an = 1 + 1 2 + · · · + 1 n (b) an = Fn+1 Fn where Fn is the n th Fibonacci Number. 2