LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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FIRST SEMESTER – April 2009
ST 1808 – ANALYSIS
Date & Time: 17/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION-A (10X2=20 marks)
Answer ALL the questions.
(1)Define a metric space and distinguish between bounded and unbounded metric spaces.
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(2) Show that ρ (f, g) = ∫| f(x)- g(x)| . dx is a metric on the class of all bounded , continuous real
0
functions on [0,1].
(3) Examine if the set of all vectors (x1, x2, x3) with x1+x 2= 1 is a vector space, where x1, x2, x3 Î R.
(4) Examine whether the set {1, ½, 1/3, 1/4…} is closed.
(5) Examine if the classes of closed and open sets are mutually exclusive and exhaustive.
(6) Show that the intersection of two open sets is open.
(7) Show that every convergent sequence in a metric space is a Cauchy sequence.
(8) If Ω:X → X is defined as Ω (x)=x2 , where X=[0,1/3] , show that Ω is a contraction mapping on
[0,1/3].
(9) Prove that any continuous image of a compact space is compact.
(10)For any sequence (xn) in R, show that
lim inf (-xn) = – lim sup xn.
SECTION-B (8 X 5=40 marks)
Answer any FIVE questions. Each question carries EIGHT marks.
(11) Let X and Y be two metric spaces with ρ1 and ρ2 as the respective metrics. Show that
ρ { (x 1, x2),(y1, y2 )} = max { ρi, (xi, yi) | i=1, 2}
is a metric on the Cartesian product XxY. Further, show that if X and Y are complete, then X×Y is
also complete.
(12) Show that (a) the union of any collection of open sets is open.
(b) The intersection of any collection of closed sets is closed.
(13) Let X and Y be two metric spaces and f a mapping of X into Y. Prove that f is continuous if and only
if f -1(G) is open in X, whenever G is open in Y.
(14) Let (X, ρ) be any metric space. Let a be a fixed point of X and let the function g: X → R be defined
by the equation g(x) =ρ (a, x) for all xЄ X. Show that g is continuous on X.
(15) State and prove Banach’s fixed point theorem.
(16) Define uniform convergence. Let (X, ρ) and (Y, σ ) be two metric spaces. Let fn: X→ Y be a
sequence of functions converging uniformly to a function f: X →Y. If each fn is continuous at c,
show that f is continuous at c.
(17) State and prove Cauchy’s necessary and sufficient condition for the uniform convergence of a
sequence of functions.
(18) State and prove Dini’s theorem for a sequence of real valued functions.
SECTION-C (2×20=40 marks).
Answer any TWO questions. Each question carries TWENTY marks.
(19)(a) Prove that, if V is an inner product space, then for all x,y Є V,
IIx+y II 2+ II x-yII2 = 2[IIxII 2+ II yII2]. (4 marks)
(b) The sequences {xn}, {yn} in the normed vector space V converge to x, y respectively and the
numerical sequences { αn}, { βn } converge to α, β respectively. Show that
αn xn + βn yn → αx + βy.
Prove also that, if V possesses an inner product, then xn. yn → x. y (8marks)
(c) Prove that the metrics ρ, σ on X are equivalent if there are constant λ, μ>0 such that
λ .ρ(x, y) ≤ σ(x, y) ≤ μ. ρ(x, y) for all x, y Є X. Give an example to show that the converse is not
true. (8 marks)
(20)(a) State and establish the necessary and sufficient condition for a set F to be closed. (10 marks)
(b) Prove that the set of real numbers is complete. (10 marks)
(21) (a) Let (X, ρ) be a metric space and let E с X. Show that
(i) if E is compact, then E is bounded and closed. (6 marks)
(ii) if X is compact and E is closed , then E is compact. (6 marks)
(b) Show that a continuous function with compact domain is uniformly continuous.(8 marks)
(22)(a) State and prove the necessary and sufficient condition for a bounded real valued function
f Є R (g; a, b). (10 marks)
(b) If f1, f2 Є R (g; a, b), prove that (f1 + f2) ЄR (g; a, b) and
b b b
∫ (f1+f2) dg = ∫ f1dg + ∫ f2dg (10marks)
a a a
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