BSc Mathematics SEM VI ATKT 2018-19 ATKT Maths III Topology Of Matric Spaces 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) State and prove Bessels Inequality
   • (ii) Let f be a continuous real valued periodic function, defined on [—7,7] and having period 27. If + (a, cosnx + is the Fourier series of f on then prove that : —> f(x) as n —+ co, where = — is the partial sum of the Fourier series of f
   • (b) Attempt any Two questions: 12
   • (i) Is the series the Fourier series of a function ? Justify your answer On(t) = > + (ax cos kt + by sin kt)
   • (iii) Apply Parseval’s equality to the function f(z) =a over and deduce that
   • (iv) Define Fejer’s Kernel Prove K,,(t) = —o
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) Suppose AK compact subset of R”. Show that K is sequentially compact
   • (ii) Show that compact subset of (R”,d) where d Euclidean, is closed and bounded Give an example to show that a closed and bounded subset of a metric space is not
   • (b) Attempt any Two from the following: 12
   • (i) Prove that a subset of a discrete metric space is compact if and only if it is finite
   • (ii) (X,d) is a metric space and is a sequence in X such that (x,,) converges to some point X. S = N}U {p} then show that S is compact by using the definition of compactness
   • (iii) Let A,B be compact subsets of d), distance d being usual. Show that A x B is a compact subset. of where d’ is the Euclidean distance Paper Subject Code: 10241 Paper III Topology of Matric Spaces
   • Q. P. Code : 51415
   • (iv) Consider the metric space (R,d), where d is the usual distance . Show that {(4, 1) :n N} is an open cover of (0,1). Is (0,1) compact ? Justify your answer
3. Q3 (a) Attempt any One of the following: 8 marks
   • (i) Show that a subset C R (with respect to usual metric of R) is connected if and only if is an interval
   • (ii) Show that a metric space (X,d) is connected if and only if every continuous function
   • (b) Attempt any Two questions: 12
   • (i) Prove that a metric space (X,d) is connected if and only if for each nonempty proper
   • (ii) Let (X,d) be a discrete metric space. If A is a X having more than one element, show that A is disconnected
   • (iii) If A and B are two connected subsets of a metric space (X,d) such that AN B 0 then prove that AU B is connected
   • (iv) If (X,d) is a connected metric space and is any metric space where Y is a finite set, then show that any continuous function f —+ Y is constant
4. Q4 Attempt any Three of the following: 15 marks
   • (a) < < Find the Fourier series of f. Assuming that the Fourier series of f converges to find the sum
   • (b) If the series + (a, cosnx + b,, sinna) converges uniformly on [—7, 7] to f, prove that the Fourier series of f is (a, +
   • (c) If A, B are disjoint non-empty subsets of (R”, d), d being Euclidean distance and A is closed , B is compact then show that d(A,B) >0
   • (d) If is a family of closed subsets of IR” (distance being Euclidean) . If F;,, is bounded for some and F,, then prove that is compact Prove that a convex subset of a normed linear space is path connected
   • (f) Prove or subset R? y 4 0} of (d being Euclidean distance

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