LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
ST 1808 - ANALYSIS
Date & Time: 26/06/2007 / 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION – A
------------------Answer ALL questions.
( 10 x 2 = 20 marks)
1. Let ( X, ρ ) be a metric space. Then show that a sequence in X cannot
converge to two limits.
2. Let ρ be a metric on X. Show that σ = 2 ρ is also a metric on X.
3. Define Inner product space and give an example.
4. Define ( i ) Open set ( ii ) Closed set.
5. If the set E is complete then show that E is closed.
6. State any three properties of compact sets.
7. If a given sequence { xn } satisfies xn = O ( vn ) and a given sequence
{ yn } satisfies yn = o ( vn ) then prove that xn + yn = O ( vn ).
8. Define pointwise convergence of a sequence of functions and give an
example .
9. State Weierstrass’s M – test for the uniform convergence of a series
of real / complex valued functions.
10.Prove that the lower sum of any partition is less than or equal to the
upper sum of any partition.
SECTION – B
------------------Answer any FIVE questions.
-----------------------------------
( 5 x 8 = 40 marks )
11. Prove that ‘ c ’ is a limit point of E iff B ( c ; ε ) contains infinitely
many points of E, for all ε > 0 .
12. Prove the following :
( i ) If G in X is open then G ' is closed.
( ii ) If F in X is closed then F ' is open.
13. Let ( X, ρ ) be a metric space and a  X be fixed.
Define g : X → R1 as g ( x ) = ρ ( a , x ) , x  X.
Then show that g is continuous on X.
14. Let ( X , ρ ) , ( Y , σ ) and ( W, τ ) be metric spaces and
let h = g o f : X → W be the composition of the function f : X → Y
and g : Y → W. Show that if f is continuous at x0  X and g is
continuous at y0 = f (x0 )  Y, then h is continous at x0 .
15. Show that if { x n } is a convergent sequence in ( X , ρ ), then it is a
Cauchy sequence. Verify whether or not the converse is true.
16. Show that a compact set in a metric space is complete.
17. Establish the following relations :
( i ) O ( vn ) + o (vn ) = O ( vn )
( ii ) O ( vn ) . o ( wn ) = o ( vn . wn )
( iii ) vn ~ wn = > vn + o ( wn ) ~ w n
18. If f : X → Rn ( X  Rm ) is differentiable at ξ , 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 and σ be two metrics on X. Show that they are
equivalent if there exists positive constants λ and μ  λ ρ ≤ σ ≤ μ ρ .
Verify whether or not the converse is true.
( 12 marks )
( b ) Show that the intersection of a finite collection of open sets is
open.
( 8 marks )
20 . Let V, W be normed vector spaces. If f : V → W is linear , then
prove that the following three statements are equivalent :
( i ) f is continuous on V
( ii ) there exists x0  V at which f is continuous.
( iii ) || f (x) || / || x || is bounded for x  V - { θ }.
21. State ad prove Heine – Borel theorem regarding compact sets.
22 ( a )Give examples to show that if lim sup | un |1/n = 1 , then the series
n →α
Σ un may converge or diverge.
( 10 marks ).
( b ) If f  R ( g ; a , b ) then prove that | f |  R ( g ; a , b )
and | a ∫ b f dg | ≤
a∫
b
| f | dg .
*******
( 10 marks).