LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1800/S715 – ANALYSIS

04.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=20marks)

 

1. Let Z be the set of all integers. Construct a function form Z to Z which is not one to one and also not onto.
2. Define a metric on a non-empty set x.
3. The real valued function f on R² – is defined by f (x, y) = .  Show that

lim f (x,y) does not exist as (x, y)  (0, 0) .

4. State weirstrass’s approximation theorem.
5. If is a convergent sequence in a metric space (X, P) then prove that it is a cauchy sequence.
6. If U_n = O (1/n^k-2), for what value of k converges?
7. Define the upper limit and lower limit of a sequence.
8. Find and also the double limit of x_mn as m,n where x_mn  = .
9. Let f: R^m.  Define the linear derivative of f at .
10. From the infinite series where obtain the expansion for log (1+x).

 

SECTION-B

Answer any FIVE questions.                                                                           (5×8=40marks)

 

11. Show that the space R’ is complete.
12. State and prove cauchy’s inequality.
13. Prove that the union of any collection of open sets is open and the intersection of any collection of closed sets is closed.
14. a) Show that f_n­ (x) = x  is not uniformly convergent
15. b) Let and be metric spaces. Let the sequence f_n : converge to f uniformly on x. If C is a point at which each f_n is continuous, then show that f is continuous at C.
16. Let V, W be normed vector spaces. If the function f : V W  is linear, then show that the following statements are equivalent.
17. f is continuous on V
18. there is a point at which f is continuous.

• is bounded for

16. Examine for convergence of if
18. u_n =
19. Let (be a metric space and let f₁, f₂, …..f_n be functions on X to R^‘.  The function

f = (f₁, f₂, …..f_n)  : is given by f(x) = (f₁(x) … f_n (x).  Prove that f is continuous at x₀ if and only if f₁, f₂,…..f_n  are continuous.

18. If f : is differentiable at then prove that the linear derivative of f at  is unique.

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40marks)

 

19. a) Let (and be metric spaces.  Prove that the following condition is

necessary and sufficient for the function f :  to be continuous on X:

whenever G is open in Y, then f^-1 (G) is open in X.

1. b) Show that if is a metric on x then so is given by  (x, y) =  and P and

are equivalent.                                                                                                   (12+8)

20. a) State and prove Banach’s fixed point theorem.
21. b) State and prove Heine – Borel theorem.                 (10+10)
22. a) State and prove d’ alembert’s ratio test
23. b) Discuss the convergence of where
24. c) Discuss the convergence and absolute convergence of

(8+8+4)

22. a) Show that a necessary and sufficient condition that fis that, given

there is a dissection D of [a, b] such that S (D, f, g) – s (D, f, g) < .

1. b) If f_I, f₂ R [g i a, b] then prove that f₁ f₂ R [g i a, b]
2. c) If f R [ g i a, b] then show that      (7+7+6)

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2004

ST 1800/S 715 – ANALYSIS

03.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

1. Define a bijective function.
2. Define a metric.
3. Is the set (0,1) complete? How?
4. Define the symbols Big O and small o.
5. Let f(x) = 1   if    x is rational

 

0   if    x is irrational,   0 £  x £ 1

Is the function Riemann integrable over  [0,1]?

6. Define lim inf and lim sup of a sequence x_n.
7. Define the linear derivative of a function f: X Rⁿ;  where X  R^m.
8. Find the double limit of x_mn = and .
9. Define uniform convergence of a sequence of functions.
10. Let f(x,y) = be defined on R² – {(0,0)}.  Show that  f(x,y) does not exist.

 

SECTION – B

 

Answer any FIVE questions.                                                                                     (5 ´ 8 = 40 marks)

 

11. State and prove Cauchy’s Inequality.
12. Show that R’ is complete.
13. Show that any collection of open sets is open and any collection of closed sets is closed.
14. State and prove Banach’s fixed point theorem.
15. Let {f_n} be a sequence of real functions integrable over the finite interval [a, b]. If f_n ® f uniformly on [a, b] then show that i) f is integrable over [a, b] and  ii) .
16. State and prove Weierstrass M-Test.
17. Show that A is the upper limit of the sequence {x_n} if and only if, given Î > 0

x_n <    for all sufficiently large n

x_n >    for infinitely many n

18. Show that if f Î R [ g; a, b] then Î R [g; a,b] and .

If is R.S integrable, can you say f R.S. integrable?  Justify.                                (3+3+2)

 

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. a) State and prove Cauchy’s root test.
20. b) Discuss the convergence of the infinite series whose nth terms are
21. i)                                                                          (8+6+6)

 

20. a) Define a compact metric space. Show that a compact set in a metric space is also

complete.                                                                                                                       (5)

1. b) State and prove Heine – Borel theorem. (15)

 

21. a) State and prove a necessary and sufficient condition that the function f is Riemann –

Stieltjes interable.

1. b) If f is continuous then show that f Î R [g; a,b]
2. c) If f₁, f₂ Î R [g; a,b] then show that f₁ f₂ Î R [g; a,b]                           (6+6+8]

 

22. a) Let (X, r) and Y, s) be metric spaces. Show that the following condition is necessary

and sufficient for the function f: X ® Y to be continuous on X: whenever G is open in

Y, then f^-1 (G) is open on X.

1. b) Let V,W be normed vector spaces. If the function f: V ® W is linear, then show that

the following three statements are equivalent.

1. f is continuous on V
2. There is a point at which f is continuous.

• is bounded for x V – {0}.

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 25

FIRST SEMESTER – APRIL 2006

                                                                    ST 1808 – ANALYSIS

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

SECTION- A

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

1. Define a discrete metric.  Show that it satisfies the properties of a metric.

2. If x_n →  x    and   x_n→  y   as n  → ∞, show that  x =

1. Define a norm on a vector space and give an example.
2. For all x, y є R⁽ⁿ⁾,  show that   x . y  =   is an inner product.
3. Check whether or not all the points of any open ball B( a ;  r ) are the interior points of B( a ;  r ).
4. Illustrate that an infinite union of closed  sets is not closed.
5. If  f  is continuous, one-to-one and onto function, then show that  f  ^-1 in general is not continuous.
6. Show that pointwise convergence does not imply uniform convergence of a sequence of functions.
7. Let  f( x )  =  x , 0 ≤  x  ≤ 1 .  Let D be the partition {0 ,¼ , ½ , ¾ ,1 } of   [ 0 , 1 ] .  Find  the upper sum  U( f ; D ) and the  lower sum

L( f ; D ) of the function  f( x ).

1. Let R ( g ; a , b ) be the collection of Riemann – Stieltjes integralble

functions with respect to  g on [ a , b ] .  If    f  є R (g ; a , b ),

show that   kf  є R (g ; a , b ) , where k is any constant.

SECTION – B

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

11. In B[ 1 , 2 ],  with  ρ( f , g ) =   sup  | f(x) – g(x) | ,

1≤ x ≤ 2

let  f­ _n  be given by   f _n(x) = ( 1 + x ⁿ ) ^{1 / n}      (1≤ x ≤ 2) .

Show that  f _n → f   where f(x) = x  (1≤ x ≤ 2).

12. In a metric space  (X , ρ ), if  x_n →  x    and   y_n→  y   as n  → ∞,

show that  ρ( x_n , y_n )   → ρ( x , y )  as  n  → ∞.

13. If   V is an inner product  space, prove that

║ x + y ║²  +  ║ x – y ║²  = 2 [║ x  ║²  +  ║ y ║²  ]  for all x , y  є V.

14. State three equivalent conditions for a point c є X to be a limit point of E С  X .
15. Show that every convergent sequence in a metric space is a cauchy sequence. Check whether or not the converse is true.
16. State and prove Banach’s fixed point principle.
17. Prove that a continuous function with compact domain is uniformly continuous.
18. State and prove Cauchy’s root test for the absolute convergence or divergence of a series of complex terms.

 

SECTION – C

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

19(a)  With  respect to the usual metric , prove that joint convergence of a

sequence  is equivalent to the marginal convergence of the

components of that sequence.  (10)

19(b)  State and prove Cauchy – Schwartz   inequality regarding inner

product space.   (10)

20(a)  Let V , W be the normed vector spaces.  Let f : V → W be a  linear

transformation.  Then prove that the following three statements are

equivalent :                                                               (16)

• f is continuous on V.
• There exists a point x_o in V at which f is continuous.
• ║f(x)║ ∕ ║x║ is bounded for x є  V – { ө }.

What do we conclude from the equivalence of statements (i) & (ii)?

20(b)  Show that a compact set in a metric space is complete.   (4)

21(a)  Prove that the number Λ is the upper limit of  the sequence

{x _n , n ≥ 1 } iff  for all  є > 0

• x _n < Λ + є for all sufficiently large n   and
• x _n  > Λ – є for infinitely many n.   (10)

21(b) Let  {f _n} be a sequence of real functions integrable over the

finite interval [a , b]. If f _n→ f  uniformly on [a , b],

prove that f is integrable over [a , b].  (10)

22(a) State and prove the Cauchy’s general principle of uniform

convergence of a  sequence of real or complex valued functions.  (8)

22(b) State and prove a necessary and sufficient condition for a function

f(x) to be Riemann – Stieltjes integrable on [a , b].  (12)

 

 

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

AB 17

FIRST SEMESTER – NOV 2006

         ST 1808 – ANALYSIS

 

 

Date & Time : 26-10-2006/1.00-4.00           Dept. No.                                                       Max. : 100 Marks

 

 

.SECTION – A

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

 

• Define a metric and give an example.

 

• Let ρ be a metric on X. Define σ = 2ρ. Show that ρ and σ are equivalent.

 

• Define Norm on a Vector Space. Give two examples.

 

• Write two equivalent definitions of a limit point of a set.

 

• Explain Linear function with an example.

 

• Define a contraction mapping and verify whether a contraction mapping is continuous .

 

• Suppose { x_n }   and  { v_n  }  are sequences in R¹. State the conditions under which we can write

( i ) x_n  = O ( v_n  )     ( ii )  x_n  = o ( v_n  ).

 

• State D’Alembert’s ratio test regarding convergence of a series.

 

1. State the general principle of uniform convergence of a sequence of real / complex valued functions.

 

1.  Let D₁ be any partition of [ a , b ]. If D is the partition containing all the points of division of D₁ , then show that the lower sums  satisfy the inequality               s (D , f , g )  ≥   s ( D₁ ,f , g ).

 

SECTION – B

 

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

———————————-

1. State and prove Cauchy –  Schwartz inequality regarding inner product.
2.  Prove that a necessary and sufficient condition for the set F to be closed is that  lim x_n  Π F whenever { x _n } is a convergent sequence of points in F.

ⁿ

 

1.  Let  X = R² , E = R²  – { (0,0) } and Y = R¹ .

Define g : E →    R¹  as

 

g ( x , y ) = x ³  / ( x ²  +  y ² ) ,  (x , y ) Î E

 

Show that g ( x , y )  →  0 as  ( x , y )  →  (  0 , 0 ).

 

 

1. Prove that pointwise convergence does not imply uniform

convergence of a sequence { f_n } of functions.

 

15. Prove that a linear function f : R^m → Rⁿ  is everywhere continuous.

 

1. Show that  R¹   with usual metric is complete.

 

1. Establish the following relations :

 

( i )  O ( v_n )  +  O ( w_n ) =  O ( v_n  +  w_n  )

( ii ) O ( v_n )  +  O (v_n  ) =  O ( v_n )

( i )  O ( v_n ) O ( w_n ) =  O ( v_n w_n  )

 

1. Let f : X →  Rⁿ  ( X  C R^m  ) be differentiable at ξ  Î  X. Then show that all the partial derivatives D_i f_j (ξ ) ,  i = 1,2, . . . , m ; j = 1,2, . . . , n exist and obtain the linear derivative Df (ξ ).

 

SECTION –  C

 

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

———————————–

 

19. ( a ) Let X =  R^2.  Take  x_n  = (   3n / (2n + 1) , 2n²  / (n²  – 2 ) ) ,

n = 1, 2, 3,  . . . .

Show that ( i )  x _n –|→   ( 1/2  , 2 ) as n  →  ∞

( ii ) x _n  →    ( 3/2  , 2 ) as n  → ∞

( 8 marks)

 

( b ) Let ρ  be a metric on X. Define  σ  =   ρ / ( 1 + ρ )

show that ( i )   σ  is a metric

( ii )   ρ and σ  are equivalent.     ( 12 marks)

 

20. Let ( X , ρ )  be a metric space and let  f _i  ,  i = 1,2, … , n be

functions form X to R¹ .

Define f = ( f ₁  , … , f_n ) : X →  Rⁿ   as

f ( x ) = ( f ₁( x ), . . . , f _n( x ) ). Then show that f is continuous

at  x₀  Î  X  iff  f is continuous at  x_{0 ,} for all  i  = 1, 2, 3, … , n.

 

21. ( a ) State and prove Banach’s fixed point theorem ( 16 marks)

 

( b ) State any two properties of compact sets.            ( 4  marks)

 

 

22. ( a ) State and prove Cauchy’s root test regarding convergence of series of compex terms. ( 10 marks )

 

( b ) State and prove Darboux theorem regarding Riemann – Stieltje’s integral.

( 10 marks )

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 31

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

    ST 1808 – ANALYSIS

 

 

 

Date : 28/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

——————-

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

1. Define  a metric space and give an example.
2. Let ( X , ρ) be a metric space and let Y С X  Define  σ : Y x Y → R¹ as  σ(x ,y)  =  ρ(x ,y)             x ,y   Y.  Show that (Y, σ ) is a metric subspace of (X , ρ ).
3. Let X = R² .  Take   x _n    =  (  n/ (2n+1)  ,   2n² / (n² – 2)  ) ; n = 1,2,…_.                                     Show that lim _n→∞ x _n    =  (  ½  ,  2 ) .
4. Let  V  =  B [ a , b ] be the class of bounded functions defined on        [ a , b ] .  Examine whether  sup _{a ≤ x ≤ b}  ‌‌‌| f( x) | is a norm on V .
5. Define a linear function and give an example.
6. Show that every convergent sequence in  (X, ρ )  is a Cauchy sequence.  Is the converse true?
7. State any three properties of compact sets.
8. Prove the following relations :

( i )  O ( v _n )  +  o (  v _n )  =   O ( v _n )

(ii)   O ( v _n )  .  o (  w _n )  =  o ( v _n w _n )

 

1. Apply Weierstrass’s  M – test to show that

^p  converges uniformly on  ( -∞ , ∞ ) , whenever  p > 1.

1.  Give an example of a function  f   not in  R( g ; a , b) whenever g is a non-constant function.

SECTION – B

——————-

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

 

11. State and prove Cauchy – Schwartz inequality regarding inner product on a vector space.
12. Prove that  ‛c’  is a limit point of E  iff     a sequence x _n  E  э   

x _n  ≠  c and  x _n  → c  as n →∞.

 

13. Prove the following:

( i  ) The union of any collection of open sets is open  .

( ii ) The intersection of any collection of closed sets is closed.

14. Let ( X , ρ) and ( Y, ρ) be the metric spaces. Prove that a necessary

and sufficient condition for f : X → Y to be continuous at ‛ x₀’ X is

that    f (x _n ) → f ( x₀ ) as n →∞.

 

15. Prove that a linear function f : R^m → Rⁿ is everywhere continuous.

 

16. State and prove Heine – Borel theorem regarding compact sets.

 

17. State and prove Cauchy`s root test regarding convergence of a series of complex terms.

 

18. Let f : X → Rⁿ ( X С R^m  ) be differentiable at  ξ  X. Then show that the linear derivative of     f at ξ  is unique.

 

SECTION – C

——————-

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

 

19. ( a ) Let ρ   be a metric on   X.  Define σ   =  ρ / ( 1 + ρ   ) .

Show that  ( i  )   σ  is a metric

( ii )   ρ  and  σ   are equivalent                         ( 10)

( b ) State and prove  a necessary and sufficient condition for the set

F to be closed.                                                                    (10)

 

20. ( a ) Suppose f : ( X , ρ) → ( Y , σ ) is continuous on X. Let ρ₁ be a

metric on X and  σ₁  be a metric on  Y э

( i  )  ρ  and ρ₁ are equivalent.

( ii )_. σ and σ₁ are  equivalent.

Then show that f is continuous with respect to ρ₁ and σ₁. ( 10 )

( b ) Prove that a necessary and  sufficient  for f : ( X , ρ) → ( Y , σ )

to be continuous on X is that f ^-1 (G) is open in X whenever G is

open in Y.                                                                         (10)

 

21. ( a ) Show that R¹ with usual metric is complete. ( 10 )

( b ) Find all values of  x for which the series  ∑ x ⁿ  /  n ^x

converges.                                                                   (10)

 

22. ( a ) State and prove Darboux theorem  regarding

Riemann – Stieltje’s  integral.                                   (10)

( b ) Let f : X  → Rⁿ  ( X С R^m  ) be differentiable at  ξ  X.

Then show that all the partial derivatives D_i f_j (ξ ) ,

i = 1,2,…, m ;  j = 1,2,…,n exist and obtain the linear

derivative D f (ξ ).                                                      (10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 19

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1808 – ANALYSIS

 

 

Date : 04-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section-A                                                        (10X2=20 marks)

Answer ALL the questions.

 

(1)  Show that    ρ(X₁, X₂) = E| X₁-X₂| is a metric on the space of all random variables defined on a

probability space.

(2)  If the inner product x.y of a vector y with any vector x is zero, show that y is a null vector.

(3)  If in a metric space, x_n→ x as n→ ∞, show that every subsequence of   {X_n, n≥1} converges to x.

(4)  Show that in a metric space with at least two points, all finite sets are closed.

(5)  Prove that in any metric space(X, ρ), both X and the empty set ф are open.

(6)  Give an example of a bijective continuous function, whose inverse is not continuous.

(7)  Examine whether a closed sub – space of a complete metric space is complete.

(8)  Explain the symbols “O” and “o “.

(9)  Let f: X→R⁽ⁿ⁾ , XсR⁽ⁿ⁾  . If f is differentiable at a, show that f is continuous at a.

(10) Let (X, ρ) be any metric space. Show that a contraction mapping is continuous on X.

 

Section-B                                                        (8X5= 40 marks)

 

Answer any FIVE questions. Each question carries EIGHT marks.

 

(11)  Show that if ρ is a metric on X, then so is σ given by

σ (x,y)= ρ(x,y)

 

1+ ρ(x,y)

and that ρ and σ are   equivalent metrics.

(12) Show that the composition of two continuous functions is continuous.

(13)  Prove that the space R with its usual metric is complete.

(14)  State and prove Banach’s fixed point theorem.

(15)  Show that a metric space is compact if and only if every sequence of points in X has a subsequence

converging to a point in X.

(16)  State and prove Dini’s theorem for a sequence of real valued functions.

(17)  If  f Є R(g ; a, b) on[ a, b] , show that  |f| Є R(g ; a, b) on  [a, b ] and

b                     b

|∫ f dg | ≤   ∫| f |dg

a                     a

(18)  If f is continuous on [a, b] show that f Є R (g; a, b).

 

 

 

 

 

 

Section-C                                            (2 X 20 = 40 marks).

 

Answer any TWO questions. Each question carries 20 marks

 

(19) (a)  Show that a sequence of points in any metric space cannot converge to two distinct limits.

(6 marks)

(b)  Give an example of a normed vector space, which is not an inner product space.                 (8 marks)

(c)   State and prove Cauchy –Schwartz inequality.                                                                      (6 marks)

 

 

(20)   (a)  Let (X, ρ)  and (Y, σ) be the metric spaces and let f:X → Y. Prove that f is continuous on X if and

only if f^-1 (G) is open in X whenever G is open in Y.                                                 (10 marks)

(b)  Let G be an open subset of the metric space X. Prove that G ‘=X-G is closed. Conversely, if F is a

closed subset of X,  prove that  F’ = X-F   is open.                                                            (10marks)

 

(21)  (a)  Define uniform convergence. Let (X, ρ) and (Y, σ) be two metric spaces. Let f _n: X → Y be a

sequence of functions  converging  uniformly  to a function  f:X →Y. If each f _n is continuous at c,

show that f is also continuous at c .                                                                                      (10 marks)

(b)  State and prove Weirstrass M- test for absolute convergence and uniform convergence.

(10 marks)

(22)  (a)  What is meant by Riemann – Stieltjes integral? Establish the necessary and sufficient condition

for a bounded real valued function f Є R(g ; a, b).                                                              (12 marks)

(b)  If f is a continuous function on [a, b], show that there exists a number c lying between a and b

such that

b

∫ f dg  =    f(c) [g(b)-g(a)].                                                     (8 marks)

a

 

 

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   LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 31

FIRST SEMESTER – April 2009

ST 1808 – ANALYSIS

 

 

 

Date & Time: 17/04/2009 / 1:00 – 4:00 Dept. No.                                                     Max. : 100 Marks

 

SECTION-A (10X2=20 marks)

                                                                    Answer ALL the questions.

 

(1)Define a metric space and distinguish between bounded and unbounded metric spaces.

1

(2) Show that ρ (f, g) =    ∫| f(x)- g(x)| . dx is a metric on the class of all bounded , continuous real

0

functions on  [0,1].

(3)  Examine if the set of all vectors (x₁, x₂, x₃) with x₁+x ₂= 1 is a vector space, where x₁, x₂, x₃  Î R.

(4)  Examine whether the set {1, ½, 1/3, 1/4…} is closed.

(5)  Examine if the classes of closed and open sets are mutually exclusive and exhaustive.

(6) Show that the intersection of two open sets is open.

(7)  Show that every convergent sequence in a metric space is a Cauchy sequence.

(8)  If  Ω:X →  X is defined  as Ω (x)=x² , where X=[0,1/3] , show that Ω  is a contraction mapping  on

[0,1/3].

(9)  Prove that any continuous image of a compact space is compact.

(10)For any sequence (x_n) in R, show that

lim inf (-x_n) =  – lim sup x_n.

 

SECTION-B (8 X 5=40 marks)

Answer any FIVE questions. Each question carries EIGHT marks.

(11)  Let X and Y be two metric spaces with ρ₁ and ρ₂ as the respective metrics. Show that

ρ { (x ₁, x₂),(y₁, y₂ )} = max { ρ_i, (x_i, y_i) | i=1, 2}

is a metric on the Cartesian  product XxY.  Further, show that if X and Y are complete, then X×Y is

also complete.

(12)  Show that (a) the union of any collection of open sets is open.

(b) The intersection of any collection of closed sets is closed.

(13)  Let X and Y be two metric spaces and f a mapping of X into Y. Prove that f is continuous if and only

if f ^-1(G) is open in X, whenever G is open in Y.

(14)  Let (X, ρ) be any metric space. Let a be a fixed point of X and let the function g: X → R be defined

by the equation g(x) =ρ (a, x) for all xЄ X. Show that  g is continuous on X.

(15)  State and prove Banach’s fixed point theorem.

(16)  Define uniform convergence. Let (X, ρ) and (Y, σ ) be two metric spaces.  Let f_n: X→ Y be a

sequence of functions converging uniformly to a function f: X →Y. If each f_n is continuous at c,

show  that f is continuous at c.

(17) State and prove Cauchy’s necessary and sufficient condition for the uniform convergence of a

sequence of functions.

(18)  State and prove Dini’s theorem for a sequence of real valued functions.

 

 

SECTION-C (2×20=40 marks).

Answer any TWO questions. Each question carries TWENTY marks.

(19)(a)   Prove that, if V is an inner product space, then for all x,y Є V,

IIx+y II ²+ II x-yII² = 2[IIxII ²+ II yII²].                                             (4 marks)

(b)  The sequences {x_n}, {y_n} in the normed vector space V converge to x, y respectively and the

numerical sequences { α_n}, { β_n } converge to α, β respectively. Show that

α_n  x_n + β_n  y_n → αx + βy.

Prove also that, if V possesses an inner product, then   x_n. y_n → x. y                  (8marks)

(c)   Prove that the metrics ρ, σ on X are equivalent if there are constant λ, μ>0 such that

λ .ρ(x, y) ≤ σ(x, y) ≤ μ. ρ(x, y) for all x, y Є X. Give an example to show that the converse is not

true.                                                                                                                      (8 marks)

(20)(a)   State and establish the necessary and sufficient condition for a set F to be closed. (10 marks)

(b)  Prove that the set of real numbers is complete.                                                   (10 marks)

(21) (a)  Let (X, ρ) be a metric space and let E с X. Show that

(i)  if E is compact, then E is bounded and closed.                                                         (6 marks)

(ii) if X is compact and  E is closed , then E is compact.                                                 (6 marks)

(b)  Show that a continuous function with compact domain is uniformly continuous.(8 marks)

(22)(a)  State and prove the necessary and sufficient condition for a bounded real valued function

f Є R (g; a, b).                                                                                                         (10 marks)

(b)   If f₁, f₂ Є R (g; a, b), prove that (f₁ + f₂) ЄR (g; a, b) and

b                               b                b

∫ (f₁+f₂) dg =    ∫ f₁dg + ∫ f₂dg                                                            (10marks)

a                               a                a

 

 

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