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

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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

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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

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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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