Metric Spaces — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. Part of the questions to be attempted together. 1. (a) (i) Let X = R and for x, y e R, define d(x, y) by JIx-y|+1, if exactly one among x and y is strictly positive d(x, y) = 1x l, otherwise Prove that (X, d) is a metric space. (4) (ii) Let X denote the set of all Riemann integrable functions defined on [a, b]. For f, g e X, define d(f, g) = S°l (x) - g(x)| dx. Show that (X, d) is not a metric space. (3.5) (b) Let X be the set of all continuous functions defined on [0, 2] equipped with metrics d(f, B) = SuP xe102|f(x) - g(x)| and d, (f, g) = J.lr(x) -g(x)/ dx, f, g e X. Compute the distance d(f, g) and d, (f, g) where sin x, 0≤x < -f(x) = and g(x) = (c) (i Define complete metric space. Let X = N the set of natural numbers. Define d(m, n) = ; m, n eX. Show that (X, d) is an incomplete metric space. (4) (ii) Is the metric space (X, d) of the set X of rational numbers with usual metric d a complete metric space? Justify your answer. (3.5) 2. (a) (i) Define an open set in a metric space (X, d). Show that a subset G in a metric space (X, d) is open if and only if it is the union of all open balls contained in G (5) (ii) Let S(x, r) be an open ball in a metric space (X, d). Let A be a subset of X such that diameter of A, d(A) < r and S(x, r) nA # . Show that A c S(x, 2r). (2.5) (b) (i) Let (X, d) be a metric space. Prove that the closed ball S(x,r), where x e X and r > O, is a closed subset of X. (3.5) (ii) Let X = R with usual metric. Let F = (1, 1/2, 1/3,...). Find the derived set of F. (4) (c) Let (X, d) be a metric space and A, and A, be subsets of X. Prove that Is the closure of the union of an arbitrary family of the subsets of X equal to the union of the closures of the members of the family? Justify your answer. (7.5) 3. (a) Let (X, d) be a metric space and Z g Ye X. If clyZ and clyZ denotes the closures of Z in the metric spaces X and Y respectively, then show that cy<= Ynclyz. (7.5) (b) Let (X, dx) and (Y, dy) be metric spaces and let f: X → Y be a function. Prove that the following statements are equivalent: (i) F(B) = f'(B) for all BEY. (7.5) (c) Let (X, dy) and (Y, d,) be metric spaces. When will a function f: X → Y is said to be uniformly continuous? For any metric space (X, d) and A € X, let g: X → R be defined as g(x) = inf{d(x,y) : y e A}. Prove that g is uniformly continuous mapping using s-5 method. (7.5) 4. (a) Define equivalent metrics, Show that the metrics d,, d,, do defined on R" by d, (x, y) = i=l d(x, y) - max {|x, - y: 1 ≤ i ≤ n) are equivalent metrics. (7.5) (b) Define fixed point of the mapping T. Show that if T: X → X is a mapping and T* has a fixed point x, e X, then T also has a fixed point. Is the converse true? Justify your answer. (7.5)