## Loyola College M.Sc. Mathematics Nov 2006 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      M.Sc. DEGREE EXAMINATION – MATHEMATICS

 AA 19

FIRST SEMESTER – NOV 2006

# MT 1805 – REAL ANALYSIS

Date & Time : 28-10-2006/1.00-4.00         Dept. No.                                                       Max. : 100 Marks

1. a)(1) When does the Riemann-Stieltjes integral reduce to Riemann integral. Explain with usual notations.

OR

(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s).                                                                                                             (5)

b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a).                                                                                                                                                (8)

(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and .                                                                                                     (7)

OR

(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)

(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x o Î (a,b) then prove that F is differentiable at x o and F¢ ( x o ) = f (x o ).                                                                           (7)

1. a) Let : [a,b] ® R m and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.

OR

(2) Suppose that  maps a convex open set E Í Rn into Rm,  is differentiable on E and there exists a constant M such that  M, ” x Î E, then prove that

ú  (b) –  (a)ú £ M ú b – aú , ” a, b Î E.                                                                      (5)

1. b) (1) Suppose E is an open set in R n ; maps R into R m ; is differentiable at x o Î E,  maps an open set containing    (E) into R k and  is differentiable at f (xo). Then the mapping of E into R k, defined by is differentiable at xo and .                                                                                                  (8)

(2) Suppose  maps an open set EÍ Â n into Â m. Let   be differentiable at x Î E, then prove that the partial derivatives (Dj f i) (x) exist and , 1£ j £ m, where {e 1, e  2, e  3, …, e n} and {u 1, u 2, u 3, …, u m} are standard bases of R n and R m.  (7)

(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x.                                          (15)

III.  a) (1) Prove:  where {f n} converges uniformly to a function f on E and x is a limit point of a metric space E.

OR

(2) Suppose that {f n} is a sequence of functions defined on E and suppose that                  ½f n (x)½£ M n, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges.                                                                                                                (5)

1. b) (1) Suppose that K is a compact set and

* {f n} is a sequence of continuous functions on K

** {f n} converges point wise to a continuous function f on K

*** f n (x) ³ f n+1 (x), ” n ÎK, n= 1,2,… then prove that f n ® f  uniformly on K. (7)

(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E.                                                                                                                         (8)

OR

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

IV a) (1)Show that  converges if and only if n >0.

OR

(2) Prove that G  = .                                                                                         (5)

b)(1) Derive the relation between Beta and Gamma functions.                                       (7)

(2) State and prove Stirling’s formula.                                                                          (8)

OR

3) If f is a positive function on (0,¥) such that f (x+1) = x f (x);  f (1) =1 and log f is convex then prove that f (x) = G (x).                                                                                               (8)

(4) If x >0 and y >0  then                                        (7)

1. a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x 0, x 1, ..,x n more than (m+1) times then prove that there exists exactly one polynomial Pn (x) of degree £ n which  agrees with f (x) at x 0, x 1, …, x n.

OR

2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)-P(x)½£ .                                                                    (5)

b)(1) Let x0, x1, …, xn be n+1 distinct points in the domain of a function f and let P be the interpolation 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 x 0, x 1, …, x n  and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that  where A (x) = (x – x0) (x – x1)…(x – x n).            (7)

(2) Let P n+1 (x)= x n+1 +Q(x) where Q is a polynomial of degree £ n and let maximum of ½P n+1 (x)½, -1 £ x £ 1.  Then prove that we get the inequality . Moreover , prove that   if and only if , where T n+1 is the Chebyshev polynomial of degree n+1.                                          (8)

OR

3) 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 the approximation and let D = . Then prove that

(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).

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

 CV 28

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.

1. (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)

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.

(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)

1. (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)

1. (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)

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.

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

# XZ 26

FIRST SEMESTER – APRIL 2008

# MT 1805 – REAL ANALYSIS

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)

OR

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

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)

1. 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)

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)

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

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)

OR

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

1. 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)

1. 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

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)

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AB 27

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

# MT 1805 – REAL ANALYSIS

Date : 06-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

I a)1) If  (with the usual notations) holds for some P and some prove that the same holds for every refinement of P.

OR

2) If f is continuous on [a,b] then prove that   on [a,b].                                     (5)

1. b) Suppose on [a,b], , is continuous on [m,M], and h(x) = (f(x)) on [a,b]. Then prove that on [a,b]
2. c) State and prove the fundamental theorem of Calculus for a function on [a,b]. (9 + 6)

OR

1. d) Let be a monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real function on [a,b] then prove that on [a,b] on [a,b].              In this case
2. e) If f maps [a,b] into Rk and if for some monotonically increasing function on [a,b]  then prove that  and                                               (8+7)
3. II. a) 1) Let be the set of all invertible operators on .Then prove that is open and the mapping A A-1 is continuous on .

OR

2) Let f be a differentiable function from E into Rm where E is an open set contained in Rn. Then prove that the linear transformation from Rn to Rm is unique.                                  (5)

1. b) Define Convex set and prove: Suppose that  maps a convex set E  into ; is differentiable on E and there exists a constant M such that then . Also prove that if f’(x) = 0 for all x in E then f is constant.
2. c) State and prove the chain rule on the differentiability of a function.         (7+8)

OR

1. d) Suppose that maps a convex set E into . Let  is differentiable at x Then prove that the partial derivatives Dj fi (x) exists and , where {e1, e2, …, en} and  {u1, u2, …, um} are the standard basis of  and respectively.                                                                     (15)

III.a) 1) Let denote the set of all continuous, complex valued, bounded functions on X. prove that is a complete metric space.

OR

2) If  is a sequence of continuous functions on E, and if uniformly on E, then prove that f is continuous on E. Is the converse true? Justify your answers.

1. b) State and prove the Weierstrass approximation theorem. (15)

OR

1. c) Let be monotonically increasing on [a,b]. Suppose on [a,b], for n = 1,2,3,… , and suppose uniformly on [a,b], Then prove that on [a,b] and
2. d) Define equicontinuity of a function and prove: If K is compact, if for n = 1,2,3,… and if {fn} is pointwise bounded and equicontinuous on K, then

(i)   {fn}  is uniformly bounded on K,

(ii)  {fn}  contains a uniformly convergent subsequence.                                                       (6+9)

1. a)1) Prove that G = .

OR

2) If then prove that where E is a periodic function with period 2.       (5)

1. b) Define Gamma function and derive a simple approximate expression for when x takes on very large values.
2. c) Derive the relationship between Beta and Gamma function. (10+5)

OR

1. d) Explain with usual notations: Fourier series, orthogonal and orthonormal system. And prove the following theorem: Let {fn } be orthonormal on [a,b]. Let S n (x) = be the nth partial sum of the Fourier series of f and suppose that tn (x) = . Then prove that  and equality holds if and only if gm =  c m , m = 1,2, …,n.           (15)
2. V) a) 1) If f has a derivative of order n at a point x0, then prove that the Taylor Polynomial is the unique polynomial such that whatever Q may be in P ( n ).

OR

2) Define Chebyshev polynomial and list down its properties.                                              (5)

1. b) Given n+1 distinct points x 0,x 1, …, x n and n+1 real numbers f (x0), f (x1),  …,       f (x n) not necessarily distinct, then prove that there exists one and only one polynomial P of degree £ n such that P (x j) = f (x j) for each j = 0,1,2,…,n.  and the polynomial is given by the formula  where .
2. c) Let P n+1 (x)= x n+1 + Q(x) where Q is a polynomial of degree £ n and let maximum of ½P n+1 (x)½, -1 £ x £ Then prove that we get the inequality . Moreover , prove that  if and only if , where T n+1 is the Chebyshev polynomial of degree n+1.                                                                                                                               (7 + 8)

OR

1. c) 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 the approximation and let D = . Then prove that

(a) If D = 0 the function R is identically zero on [a,b].

(b) If D > 0, the function R has at least (n+1) changes of sign on [a,b].                               (15)

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## Loyola College M.Sc. Mathematics Nov 2012 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

# MT 1816 – REAL ANALYSIS

(12 BATCH STUDENTS ONLY)

Date : 05/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions. Each question carries 20 marks.

I.a) 1) Prove that refinement over an interval increases the lower sum and decreases the   upper sum.

OR

a)2. Using the notion of Upper sums and Lower sums of a bounded function, when do you say that  on [a,b]. When does the Riemann-Stieltjes integral reduces to Riemann integral?                                                                                      (5)

b)1) Suppose f is bounded on [a,b]. f has finitely many points of discontinuity on [a,b] and is continuous at every point at which f is discontinuous the prove that  .

b)2) Prove: Let  be a monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real function on [a,b] then prove that on [a,b] on [a,b]. In this case                                                                                   (5 +10)

OR

c)1) Prove: on [a,b]

c)2) Suppose c Î (a,b) and two of the three integrals ,  and  exist. Then prove that the third also exists and

c)3) If f is monotonic on [a,b] and if   is continuous on [a,b], then  prove that .                                   (4+4+7)

1. a)1) State and prove the theorem on Cauchy criterion for uniform convergence.

OR

a)2) Using a suitable example show that the limit of the integral need not be equal to the integral of the limit.                                                                                           (5)

b)1) If X is a metric space and denote the set of all complex valued, continuous, bounded functions with domain X. Choosing a suitable distance function between the elements of   prove that is a metric space  and also a complete metric space.

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

OR

c)1) Let . Verify whether .

c)2) Suppose  converges uniformly to a function f on E, where E is a set as a metric space. Let x be a limit point of  E and suppose that . Then prove that {An}conveges and                                                               (5+10)

III. a)1) State and prove Parseval’s formula.

OR

a)2) State and prove Dini’s theorem.                                                                          (5)

b)1) State and prove Jordan’s theorem.

b)2)State and prove Riemann- Lebesgue Lemma.                                                   (6+9)

OR

c)1) State and prove Riesz – Fischer’s theorem.

c)2) State and prove Riemann Localization theorem.                                                   (6+9)

1. a)1) Suppose E is an open set in Rn and f maps E into Rm and x is an element in E such that when , . Then prove that A is unique.

OR

a)2) Let be the set of all invertible linear operators on Rn. If then prove that .                                                                                                                                                                           (5)

1. b) State and prove Inverse function theorem.                           (15)

OR

c)1) If  and c is a scalarthen prove that  . And with the distance between A and B defined as   , prove that is a metric space.

c)2) Define Contraction principle and prove the following theorem: Let X is a complete metric space and if  is a contraction of  X into X then  there exists only one xX  such that (x)=x.                                                                                                                        (5+10)

Va)1). Derive the rectilinear co ordinates.

OR

a)2) Derive the sum of powers of .                                                                         (5)

b)1) Explain how the product and quotient rule are derived for functions f(x) and g(x)?

OR

b)2) Derive the expression for D’ Alembert’s wave equation for a vibrating string.

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

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

 CV 13

DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2007

MT 5501REAL ANALYSIS

Date & Time: 28/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

Answer all the questions:                                                                           10 x 2 = 10

1. Define an inductive set with an example.

1. Prove that every positive integer n (except 1) is either a prime or a product of primes.

1. State and prove Euler’s theorem for real numbers.

1. Define a Metric space.

1. State Cantor’s intersection theorem for closed sets.

1. Define an interior point and an open set.

7.Give an example of a uniformly continuous function.

1. Define a Cauchy sequence.

1. Suppose f and g are defined on (a, b) and are both differentiable at c Î (a, b), then prove

that the function fg is also differentiable at c.

1. Define total variation of a function f on .

Answer any five questions:                                                                                         5 x 8=40

1. Prove that the set R of all real numbers is uncountable.

1. State and prove Bolzano-Weirstass theorem for R.

1. Prove that every compact subset of a metric space is complete.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. If X is compact, then prove that f (X) is a compact subset of Y.

1. Let (X, d1) and (Y, d2) be metric spaces and f: X Y be continuous on X. Then show that a map f: X Y is continuous on X if and only if f -1 (G) is open in X for every open set G in Y.

16    Prove that in a metric space (X, d)

( i ) Arbitrary union of open sets in X is open in X

( ii) Arbitrary intersection of closed sets in X  is closed in X.

1. Let f: R and f have a local maximum or a local minimum at a point c.

Then prove that f ’(c) = 0.

1. Let f be of bounded variation onand xÎ (a, b) Define V:  R as   follows:

V (a) = 0

V (x) =Vf , a <  x ≤ b.

Then show that the functions V and V – f are both increasing functions on.

Answer any two questions:                                                                                                      2 x 20 = 40

19   State and prove Intermediate value theorem for continuous functions.

1.   State and prove Lagrange’s theorem for a function f :  R

21.(a) Suppose c Î (a ,b) and two of the three integrals f da ,f da , and f da

exists. Then show that the third also exists andf da =f da +f da.

(b) When do we say f is Riemann-Stieltjes integrable?

1. (a) State and prove Unique factorization theorem for real numbers.

(b) If F is a countable family of countable sets then show that  is also countable.

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## Loyola College B.Sc. Mathematics Nov 2008 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# AB 19

FIFTH SEMESTER – November 2008

# MT 5501 – REAL ANALYSIS

Date : 05-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION-A

Answer ALL questions:                                                                                           (10 x 2=20 marks)

1. State principle of induction.
2. Show that the set Z is similar to N.
3. Define isolated point of a set in a metric space.
4. “Arbitrary intersection of open sets in open”  True or False. Justify your answer.
5. If {xn} is a sequence in a metricspace, show that { xn} converges to a unique point.
6. Define complete metric space and give an example of a space which is not complete.
7. When do you say that a function  has a right hand derivative at ?
8. Define (i) Strictly increasing function

(ii) Strictly decreasing function

1. When do you say that a partition is a refinement of another partition? Illustrate by an example.
2. Define limit superior and limit inferior of a sequence.

SECTION-B

Answer any FIVE questions:                                                                                   (5 x 8=40 marks)

1. Show that is an irrational number.
2. Show that collection of all sequences whose terms are 0 and 1 is uncountable.
3. Let Y be a subspace of a metric space (X,d). Show that a subset A of Y is open in Y if an only if  for some open set G in X.
4. Show that every compact subset of a metric space is complete.
5. Show that every compact set is closed and bounded in a metric space.
6. Let be differentiable at c and g be a function such that where I is some open interval containing the range of f. If g is differentiable at f(a), show that gof is differentiable at c and (gof)(c)=g(f(o).f(c).

1. If f is of bounded variation on [a,b] and if f is also of bounded variation on [a,c] and [c,b] for , show that .
2. Show that lim inf(an) if and only if for .
• there exists a positive integer N such that for all  and
• given any positive integer m, there exists such that .

SECTION-C

Answer any TWO questions:                                                                                   (2 x 20=40 marks)

1. (a) State and prove Unique factrization theorem for integers.

(b) If S is an infinite set, show that S contains a countably infinite set.

(c) Given a countable family F of sets, show that we can find a countable family G of pairwise disjoint sets such that .

1. (a) State and prove Heine theorem.

(b) If S, T be subsets of a metric space M,

show that        (i)

(ii)

Illustrate by an example that

1. (a) Let X be a compact metric space and  be continuous on X.

Show that is a compact subset of Y.

(b) Show that  on R is continuous but not uniformly continuous.

1. (a) Let f be of bounded variation on [a,b] and V be the variation of f. Show that V is continuous

from the right at if and only if f is continuous from the right at c.

(b) on [a,b] and g is strictly increasing function defined on [c,d] such that

g ([c,a])=[a,b]. Let h (y) = f (g (y)) and .

Show that  .

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## Loyola College B.Sc. Mathematics Nov 2008 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# AB 10

FIFTH SEMESTER – November 2008

# MT 5505 – REAL ANALYSIS

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer  ALL  Questions                                                                                           (10 x 2=20 marks)

1. Define similar sets with an example.
2. If a and b are any real numbers such that

1. Define a metric space.
2. Give an example of a set E in which every interior point of E is also an accumulation point of E but

not conversely.

1. Define a convergent sequence.
2. State intermediate value theorem for continuous functions.
3. Define Open ball and Closure of set E.
4. When is a sequence {an} said to be Monotonic increasing and decreasing?
5. State the linearity property of Riemann- Stieltjes integral.
6. Define limit superior of a real sequence.

SECTION – B

Answer  ANY FIVE Questions.                                                                                (5 x 8=40 marks)

1. State and prove Minkowski’s inequality.
2. If n is any positive integer, then prove that Nn is countably infinite.
3. Let(X, d ) be a metric space. Then prove that
4. i) the union of an arbitrary collection of open sets in X is open in X.
5. ii) the intersection of an arbitrary collection of closed sets in X is closed in
6. Prove that a closed subset of a compact metric space is compact.
7. Prove that every compact subset of a metric space is complete.
8. Let f : ( X, d ) → Rk be continuous on X. If X is compact, then prove that f is

bounded on X.

1. State and prove Rolle’s theorem.
2. Let {an} be a real sequence. Then prove that

(i) {an} converges to l  if and only if  lim inf an = lim sup an = l

(ii) {an} diverges to + ∞ if and only if  lim inf an = + ∞

SECTION C

Answer  ANY TWO  Questions.                                                                              (2 x 20 = 40 marks)

1. (a) Prove that the set R is uncountable. (10 marks)

(b) State and prove Cauchy –Schwartz inequality.                                                          (10 marks)

1. (a) State and prove Bolzano- Weierstrass theorem. (18 marks)

(b) Give an example of a metric space in which a closed ball

is not the  closure of the open ball B(a ; r ).                                                                 (2 marks)

1. (a) Let ( X, d1) and (Y, d2) be metric spaces and f : XY. If x0X, then prove that f is

continuous at x0 if and only if for every sequence {xn} in X that converges to x0,

the sequence { f (xn ) } converges to f (x0).                                                                  (12marks)

(b) Prove that Euclidean space k is complete.                                                                ( 8 marks)

1. b) State and prove Taylor’s theorem.                                                                  (10 marks)

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## Loyola College B.Sc. Mathematics April 2009 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 ZA 29

FIFTH SEMESTER – April 2009

# MT 5505 / 5501 – REAL ANALYSIS

Date & Time: 16/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

SECTION-A

Answer all questions:                                                                        (10 x 2=20)

1. Define an order – complete set and give an example of it.
2. When do you say that two sets are similar?
3. Define discrete metric space.
4. Give an example of a perfect set in real numbers.
5. Define open map and closed map.
6. When do you say that a function is uniformly continuous?
7. If a function f is differentiable at c, show that it is continuous.
8. Define “total variation” of a function f on [a,b].
9. Give an example of a sequence {an}whose lim inf and lim Sup exist, but the sequence is not convergent.
10. Give an example of a function which is not Riemann Stieltjes integrable.

SECTION-B

Answer any five questions:                                                              (5 x 8=40)

1. Let a,b be two integers such that (a,b)=d. Show that there exists integers  such that .
2. Show that the set of all real numbers is uncountable.
3. Let E be a subset of a metric space X. Show that  is the smallest closed set containing E.
4. State and prove Heire Borel theorem.
5. Show that in a metric space every convergent sequence is Cauchy, but not conversely.
6. Let f, g be differentiable at Show that f  g  is differentiable at c and if is also differentiable at c.
7. State and prove Lagrange’s mean valve theorem.
8. Suppose {an}is a real sequence. Show that lim Supan=l if and only if for every ,
• there exists a positive integer N such that for all and
• given any positive integer m, there exists an integer such that .

SECTION-C

Answer any two questions:                                                                          (2 x 20=40)

1. (a) State and prove Minkowski’s inequality.

(b) Show that there is a rational number between any two distinct real numbers.

(c) If show that e is irrational.

1. (a) If F is a family of open intervals that covers a closed interval [a,b], show that a finite sub family of F also covers [a,b].

(b) Let SÌRn. If every infinite subset of S has an accumulation point in S, show that S is closed and bounded.

1.  (a) Let (X,d1), (Y, d2) be metric spaces and . Show that f is continuous at if and only if for every sequence in X that converges to  the sequence converges to .

(b) State and prove Bolzano theorem.

1. (a) State and prove Taylor’s theorem.

(b) Suppose on [a,b]. Show that on [a,b] and

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## Loyola College B.Sc. Mathematics April 2011 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2011

# MT 5505/MT 5501 – REAL ANALYSIS

Date : 11-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 9:00 – 12:00

SECTION  A

Answer ALL questions.                                    (10 x 2 = 20)

1. State the least upper bound axiom.

1. Prove that any infinite set contains a countable subset.

1. Prove that the intersection of an arbitrary collection of open sets need not be open.

1. Distinguish between adherent and accumulation points.

1. Prove that any polynomial function is continuous at each point in .

1. Give an example of a continuous function which is not uniformly continuous.

1. State Rolle’s theorem.

1. If a real-valued function has a derivative at , prove that is continuous at .

1. Give an example of a sequence of real numbers whose limit inferior and limit superior exist, but the sequence is not convergent.

1. Give an example of a function which is not Riemann-Stieltjes integrable.

SECTION  B

Answer ANY FIVE questions.                                     (5 x 8 = 40)

1. State and prove Cauchy-Schwartz inequality.

1. Prove that the Cantor set is uncountable.

1. Prove that a subset E of a metric space is closed if and only if it contains all its adherent points.

1. Prove that a closed subset of a complete metric space is also complete.

1. State and prove Lagrange’s mean value theorem.

1. If a real-valued function is monotonic on , prove that the set of discontinuities of is countable.

1. If a real-valued function is continuous on , and if exists and is bounded in , prove that  is of bounded variation on .

1. State and prove integration by parts formula concerning Riemann-Stieltjes integration.

SECTION  C

Answer ANY TWO questions.                                     (2 x 20 = 40)

 19. (a) Prove that the set of rational numbers is not order-complete. (b) Prove that the set of all rational numbers is countable. (c) State and prove Minkowski’s inequality.                                                          (10+5+5) 20. (a) Prove that every bounded and infinite subset of  has at least one accumulation point. (b) State and prove the Heine-Borel theorem.                                                       (16+4) 21. (a) Let  and  be metric spaces and . Show that  is continuous at  if and only if for every sequence  in X that converges to , the sequence  converges to . (b) Prove that a continuous function defined on a compact metric space is uniformly continuous.                                                                                                        (10 + 10) 22. (a) State and prove Taylor’s theorem. (b) Prove that a monotonic sequence of real numbers is convergent if and only if it is bounded.                                                                                                                (12+8)

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## Loyola College B.Sc. Mathematics April 2012 Real Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIFTH SEMESTER – APRIL 2012

# MT 5505/MT 5501 – REAL ANALYSIS

Date : 25-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions:                                                                                                        (10 x 2 = 20)

1. State and prove the triangular inequality.

1. Prove that the sets Z and N are similar.

1. Prove that the union of an arbitrary collection of closed sets is not necessarily closed.

1. Prove that every neighbourhood of an accumulation point of a subset E of a metric space contains infinitely many points of the set E.

1. Show that every convergent sequence is a Cauchy sequence.

1. Define the term “complete metric space” with an example.

1. State Rolle’s theorem.

1. Prove that every function defined and monotonic on is of bounded variation on .

1. State the linearity property of Riemann-Stieltjes integral.

1. State the conditions under which Riemann-Stieltjes integral reduces to Riemann integral.

PART – B

Answer ANY FIVE questions:                                                                                 (5 x 8 = 40 marks)

1. State and prove Cauchy-Schwartz inequality.

1. Prove that the interval is uncountable.

1. State and prove the Heine-Borel theorem.

1. State and prove the intermediate value theorem for continuous functions.

1. Let and be metric spaces and . If  is compact and  is continuous on , prove that  is uniformly continuous on .

1. State and prove the intermediate value theorem for derivatives.

1. Suppose on . Prove that on  and that

.

1. a) Let be a real sequence. Prove that (a) converges to L if and only if

(b)  diverges to  if and only if .

PART – C

Answer ANY TWO questions:                                                                                             (2 x 20 = 40)

 19. (a) Prove that every subset of a countable set is countable. (b) Prove that countable union of countable sets is countable. (c) State and prove Minkowski’s inequality.                                                            (8+7+5) 20. (a) Prove that the only sets in R that are both open and closed are the empty set and the set R itself. (b) Let E be a subset of a metric space . Show that the closure  of E is the smallest closed set containing E. (c) Prove that a closed subset of a compact metric space is compact.                     (4+8+8) 21. (a) Let  and  be metric spaces and . Prove that  is continuous on X if and only if  is open in X for every open set G in Y. (b) Explain the classification of discontinuities of real-valued functions with examples. (12+8) 22. (a) State and prove Lagrange’s mean value theorem. (b) Suppose  on  and  for every  that is monotonic on . Prove that  must be constant on . (c) Prove that a bounded monotonic sequence of real numbers is convergent.       (8+4+8)

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