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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