BSc Mathematics SEM VI 2016 17 2016-17 Maths Paper III Matric Space Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 (a) Attempt any One question: 8 marks
   • (i) Let f be a continuous real valued periodic function, defined on and having period 27. If f ~ + (a, cosnx + b, is the Fourier series of f on 7] then prove that : — = = fle) D,,(t)dt where D,,(x) is the Dirichlet’s kernel
   • (ii) State and prove Bessels Inequality
   • (b) Attempt any Two questions: 12
   • (i) For f(z) and + (a, cosnx find Fourier On(t) = > + =| (ax, cos kt + by sin kt)
   • (iii) Define Fejer’s Kernel Prove that. =
   • (iv) Is the series the Fourier series of a function f ? Justify your answer
2. Q2 (a) Attempt any One question: 8 marks
   • (i) AK C R” (distance Euclidean), is closed and bounded. Show that K is sequentially
   • (ii) Let = x C (distance Euclidean). Prove that J is
   • (b) Attempt any Two questions: 12
   • (i) Let (X,d) and (Y,d') be metric spaces. If (X,d) is compact and f : X isa continuous function, then show that f(X) is a compact subset of Y
   • (ii) Show that {0} U is a compact subset of where d is usual distance in R using the definition of a compact subset
   • (iii) Let (X,d) be a compact metric space and f : X —> (0,00) is continuous. Show that there exists such that Va X
   • Q. P. Code: 04995
   • (iv) Prove that a subset of a discrete metric space is and only if it is finite
3. Q3 (a) Attempt any One question: 8 marks
   • (i) Show that a subset C R is connected if and only if & is an interval (distance being
   • (ii) Show that a metric space (X,d) is connected and only if every continuous function
   • (b) Attempt any Two questions: 12
   • (i) Let (X,d) be a metric space and A be a connected subset of X. If BC A, then show that B is connected. In particular, prove that A is connected
   • (ii) Show that S = R* : y £0} is not connected. Hence show that S$ not path connected in a Euclidean metric space R?
   • (iii) If (X,d) is a connected metric space and f : X —>+ Z a continuous function, prove that f is constant. (distance in Z being usual)
   • (iv) Show that a convex set in R” is path connected (distance being Euclidean)
4. Q4 Attempt any Three from the following: 15 marks
   • (a) If the series > + (a, cosnx + converges uniformly to f on then prove that Fourier series of f is + (a, + b,
   • (b) f(x) = Test Find the Fourier series of f. Assuming that the Fourier series of f converges to x = 0, find the sum
   • (c) Show closed subset of a compact set is compact in any metric space
   • (d) Show that = = 1} is a compact subset of distance being Let A and B be path connected subsets of a metric space (X,d) such that AN 0 Show that AU B is path connected
   • (f) Let (X,d) be metric space which is not bounded. Prove that for each X and each r the set X : =r} is non-empty

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