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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