LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
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FIRST SEMESTER – APRIL 2007
MT 1805 – REAL ANALYSIS
Date & Time: 27/04/2007 / 1:00 – 4:00Dept. No. Max. : 100 Marks
Answer all the questions. Each question carries 20 marks.
- (a). (i). Prove that refinement of partitions decreases the upper Riemann Stieltjes sum.
(OR)
(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)
(OR)
(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)
- (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.
(OR)
(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)
(OR)
(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)
- (a). (i).Show by means of an example that a convergent series of continuous functions
may have a discontinuous sum.
(OR)
(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)
(OR)
(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)
- (a). (i). Define the exponential function and obtain the addition formula.
(OR)
(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)
(OR)
(iii). State and prove the Parseval’s theorem. (15)
- (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.
(OR)
(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)
(OR)
(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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