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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