LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
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FIRST SEMESTER – APRIL 2008
MT 1805 – REAL ANALYSIS
Date : 30-04-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
- a) 1) Let and on [a,b] then prove that
(i) on [a,b] and (ii)
OR
2) Define step function and prove: If a < s < b, on [a,b] and , the unit step
function, then prove that (5)
- 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
OR
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)
- a) 1) Let exists then prove that it is unique.
OR
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
(5)
- 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)
OR
2) a) State and prove the Contraction principle.
- 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?
OR
- 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)
OR
3) State and prove Stone-Weierstrass theorem. (15)
- a)1) Is the trignometric series a Fourier series? Justify your answer.
OR
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)
OR
3) State and prove the Dirichlet’s necessary and sufficient condition for a Fourier series to converge to a sum s. (15)
- a)1) Write a note on Lagrange’s polynomial.
OR
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
- i) If D = 0 the function R is identically zero on [a,b].
- ii) If D>0, the function R has at least (n+1) changes of sign on [a,b]. (15)
OR
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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