Loyola College M.Sc. Mathematics April 2007 Real Analysis Question Paper PDF Download



CV 28






Date & Time: 27/04/2007 / 1:00 – 4:00Dept. No.                                              Max. : 100 Marks



Answer all the questions. Each question carries 20 marks.


  1. (a). (i). Prove that refinement of partitions decreases the upper Riemann Stieltjes sum.




(ii). If f is monotonic on [a, b], and if  is continuous on [a, b], then prove                                that  on [a, b].                                                                      (5)


(b). (i). Suppose cn ≥ 0, for n = 1, 2, 3 …, converges, and { sn} is a sequence of

distinct points in [a, b]. If  and f is continuous on [a, b], then prove that .

                  (ii). Suppose that  on [a, b], m ≤  f  ≤ M,  is continuous on   [m,M],and  on [a, b]. Then prove that  on [a, b].                                 (7+8)




(iii). Assume that  increases monotonically and  on [a, b]. Let f be a

bounded real function on [a, b]. Then prove that  if and only if  and in

that case.

(iv). State and prove the fundamental theorem of Calculus.                                            (8+7)


  1. (a). (i). Prove that a linear operator A on a finite dimensional vector space X is

one-to-one if and only if the range of A is all of X.




(ii). If  then prove that  and .               (5)


(b). (i). Let  be the set of all invertible linear operators on Rk. If

, and  then prove that .

(ii). Obtain the chain rule of differentiation for the composition of two

functions.                                                                                                                  (7+8)




                  (iii). Suppose  maps an open set E Rn into Rm. Then prove that

if and only if the partial derivatives  exist and are continuous on E for , .

(iv). If X is a complete metric space and if  is a contraction of X into X,

then prove that there exists one and only one x in X such that .                 (8+7)



  1. (a). (i).Show by means of an example that a convergent series of continuous functions

may have a discontinuous sum.


(ii). State and prove the Cauchy criterion for uniform convergence.                               (5)


(b). (i). Suppose on a set E in a metric space. Let x be a limit point of E

and suppose that . Then prove that converges and that .

(ii). Let  be monotonically increasing on [a, b]. Suppose on [a, b],

for n = 1, 2, …, and suppose that  uniformly on [a, b]. Then prove that      on [a, b] and that.                                                            (8+7)



(iii). If f is a continuous complex function on  [a, b], then prove that there

exists a sequence of polynomials Pn such that uniformly on [a, b]. (15)


  1. (a). (i). Define the exponential function and obtain the addition formula.


(ii). If , prove with usual notation that E(it) 1.                     (5)


(b). (i). Given a double sequence, i = 1,2,…,  j = 1,2,…, suppose that

and  converges. Then prove that .

(ii). Suppose that the series and converges in the segment

S = (–R, R). Let E be the set of all x in S at which  = . If E has a limit point in S, then prove that for all n.                                                                 (7+8)



            (iii). State and prove the Parseval’s theorem.                                                                   (15)








  1. (a). (i). If f has a derivative of order n at a point x0, then prove that the Taylor

polynomial  is the unique polynomial such that

for any polynomial Q of degree ≤ n.




(ii). Define the Chebychev polynomial Tn and prove that it is of degree n and that

the coefficient of xn is 2n–1.                                                                 (5)


(b). (i). State and prove the construction theorem.

(ii). Let where  is a polynomial of degree ≤ n, and let

. Then prove that , with equality if and

only if  where  is the Chebychev polynomial of degree n+1.                                                                                                                                     (8+7)



(iii). Let x0, x1, …, xn be n+1 distinct points in the domain of a function f and let P

be the interpolating polynomial of degree ≤ n, that agrees with f at these points. Choose a point x in the domain of f and let [a, b] be any closed interval containing the points x0, x1, …, xn and x. If f has derivative of order n+1 in [a, b] then prove that there is a point c in (a, b) such that , where .


(iv). If f(x) has m continuous derivatives and no point occurs in the sequence x0,

x1, …, xn more than m+1 times, then prove that there exists one polynomial  Pn(x) of degree ≤ n which agrees with f(x) at x0, x1, …, xn.                                                                                                                                                                                                      (8+7)


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