LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2008
MT 1805 - REAL ANALYSIS
Date : 30-04-08
Time : 1:00 - 4:00
Dept. No.
XZ 26
Max. : 100 Marks
I. a) 1) Let f  R( ) and g  R ( ) on [a,b] then prove that
b
b
(i) fg  R ( ) on [a,b] and (ii)  fd   f d
a
a
OR
2) Define step function and prove: If a < s < b, f  R( ) on [a,b] and  ( x)  I ( x  s ) , the unit step
b
function, then prove that

fd  f ( s )
a
(5)
b) 1) Let c  0 , n = 1,2,3,… . Suppose that  c is convergent and {sn} is a sequence of distinct
n
n

numbers in (a,b). Let  ( x)   I ( x  s ) . Let f be continuous on [a,b] then prove that
n
n 1
b

fd


 ck f (sk )

k 1
a
2) Let  be monotonically increasing function on [a,b] and let    R on [a,b]. If f is a bounded real
b
b
function on [a,b] then prove that f  R( ) on [a, b]  f    R on [a, b] and  fd   f  dx
a
a
OR
x
3) Let f  R on [a,b] for a  x  b . Define F ( x)   f (t )dt , then prove that F is continuous on [a,b].
a
If F is continuous at some point x  ( a, b) , then prove that F is differentiable at xo and
o
F ( x )  f ( x ) .
o
o
4) State and prove the fundamental theorem of Calculus and deduce the following result:
Suppose F and G are differentiable functions on [a,b], F   f  R and G  g  R then prove that
b
b
F
(
x
)
g
(
x
)
dx

F
(
b
)
G
(
b
)

F
(
a
)
G
(
a
)


 f ( x)G( x)dx
a
a
(6 + 9)
II. a) 1) Let f : E  R m , x  E. If f ( x) exists then prove that it is unique.
OR
2) Define a convex set and prove: Suppose that f maps a convex set E  R n in to R m ; f is
differentiable on E and there exists a constant M such that f ( x)  M x  E then prove that
f (a)  f (b)  M b  a a, b  E.
(5)
b) 1) When do you say a function f of E  R n is continously differentiable? Let f maps an open set
E  R n in to R m show that f   C ( E ) if and only if the partial derivatives D f exists and are
j i
continuous on E for 1  i  m and 1  j  n.
(15)
OR
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?
OR
2) If {f } is a sequence of continuous functions on E, and if f  f uniformly on E
n
n
then f is continuous on E. Is the converse true? If the converse is not true, assert
under what conditions the converse also becomes true.
(5)
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
f ( x)  lim f  ( x); a  x  b.
x  n
(5 + 10)
OR
3) State and prove Stone-Weierstrass theorem.
(15)

IV. a)1) Is the trignometric series  sin nx, a Fourier series? Justify your answer.
n 1
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 2 ) and if  0 then prove that there is a trignometric polynomial P
such that P( x)  f ( x)  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)
V. 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 D  f  T . 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
1
.

P
 max P
( x) . Then prove P
n 1
n  1 1  x  1 n  1
2n
(10+5)
2