Loyola College M.Sc. Mathematics April 2008 Real Analysis Question Paper PDF Download



XZ 26







Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00


  1. a) 1) Let and  on [a,b] then prove that

        (i) on [a,b] and (ii)


 2)   Define step function and prove: If a < s < b, on [a,b] and , the unit step

      function, then prove that                                                     (5)

  1. b) 1) Let , n = 1,2,3,… . Suppose that is convergent and {sn} is a sequence of distinct

numbers in (a,b). Let . Let  f  be continuous on [a,b] then prove that

   2) Let be monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real

       function on [a,b] then prove that


     3) Let on [a,b] for . Define , then prove that F is continuous on [a,b].

If F is continuous at some point , then prove that F is differentiable at xo and .

    4) State and prove the fundamental theorem of Calculus and deduce the following result:


Suppose F and G are differentiable functions on [a,b], then prove that                                             (6 + 9)

  1. a) 1) Let exists then prove that it is unique.


    2)   Define a convex set and prove: Suppose that  maps a convex set ;  is

          differentiable on E and there exists a constant M such  that then prove that


  1. b) 1) When do you say a function is continously differentiable? Letmaps an open set

         show that  if and only if the partial derivatives Djfi exists and are

continuous on E for                                               (15)


2) a)  State and prove the Contraction principle.


  1. b) Let C(X) denote the set of all continuous, complex valued, bounded functions onX. Prove that C(X)

          is a complete metric space.                                                             (5+10)

III. a)1) Prove that every converging sequence is a Cauchy’s sequence. Is the converse true?


  1. b) 1) State and prove the Cauchy criterion for uniform convergence.

2) Suppose {fn} is a sequence of differentiable functions on [a,b]. Suppose that {fn(x0)} converges uniformly on [a,b] then prove that {fn} converges uniformly on [a,b] to some function f and                                             (5 + 10)


3) State and prove Stone-Weierstrass theorem.                                                    (15)

  1. a)1) Is the trignometric series a Fourier series? Justify your answer.


2) Define a Gamma function and state the three properties that characterize Gamma function completely.                                                                                                            (5)

b)1) State and prove the Parseval’s theorem.

2) If f is continuous (with period ) and if  then prove that there is a trignometric polynomial P such that  for all real x.                                          (10 + 5)


3) State and prove the Dirichlet’s necessary and sufficient condition for a Fourier series to converge to a sum s.                                                                                                      (15)

  1. a)1) Write a note on Lagrange’s polynomial.


2) Write a note on Chebyshev polynomial.                                                                       (5)

b)1)  Let f be a continuous function on [a,b] and assume that T is a polynomial of degree  n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f(x) – T(x) denote the error in this approximation and let . Then prove

  1. i) If D = 0 the function R is identically zero on [a,b].
  2. ii) If D>0, the function R has at least (n+1) changes of sign on [a,b]. (15)


2)  If f(x) has m continuous derivatives and no point occurs in the sequence xo, x1, x2, …, xn more than (m + 1) times then prove that there exists exactly one polynomial Pn(x) of degree n which agrees with f(x) at xo, x1, x2, …, xn.

3) Let P n+1 (x) = x n+1+ Q(x), where Q(x) is a polynomial of degree n, and let . Then prove .                                                                              (10+5)



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