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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