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).