LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 25

FIRST SEMESTER – APRIL 2006

ST 1808 – ANALYSIS

Date & Time : 22-04-2006/1.00-4.00 P.M. Dept. No. Max. : 100 Marks

SECTION- A

Answer ALL questions . (10 x 2 = 20 marks)

Define a discrete metric. Show that it satisfies the properties of a metric.

If x_n → x and x_n→ y as n → ∞, show that x =

Define a norm on a vector space and give an example.
For all x, y є R⁽ⁿ⁾, show that x . y = is an inner product.
Check whether or not all the points of any open ball B( a ; r ) are the interior points of B( a ; r ).
Illustrate that an infinite union of closed sets is not closed.
If f is continuous, one-to-one and onto function, then show that f^-1 in general is not continuous.
Show that pointwise convergence does not imply uniform convergence of a sequence of functions.
Let f( x ) = x , 0 ≤ x ≤ 1 . Let D be the partition {0 ,¼ , ½ , ¾ ,1 } of [ 0 , 1 ] . Find the upper sum U( f ; D ) and the lower sum

L( f ; D ) of the function f( x ).

Let R ( g ; a , b ) be the collection of Riemann – Stieltjes integralble

functions with respect to g on [ a , b ] . If f є R (g ; a , b ),

show that kf є R (g ; a , b ) , where k is any constant.

SECTION – B

Answer any FIVE questions. (5 x 8 = 40 marks)

In B[ 1 , 2 ], with ρ( f , g ) = sup | f(x) – g(x) | ,

1≤ x ≤ 2

let f _n be given by f_n(x) = ( 1 + x ⁿ ) ^{1 / n} (1≤ x ≤ 2) .

Show that f_n → f where f(x) = x (1≤ x ≤ 2).

In a metric space (X , ρ ), if x_n → x and y_n→ y as n → ∞,

show that ρ( x_n, y_n) → ρ( x , y ) as n → ∞.

If V is an inner product space, prove that

║ x + y ║² + ║ x – y ║² = 2 [║ x ║² + ║ y ║² ] for all x , y є V.

State three equivalent conditions for a point c є X to be a limit point of E С X .
Show that every convergent sequence in a metric space is a cauchy sequence. Check whether or not the converse is true.
State and prove Banach’s fixed point principle.
Prove that a continuous function with compact domain is uniformly continuous.
State and prove Cauchy’s root test for the absolute convergence or divergence of a series of complex terms.

SECTION – C

Answer any TWO questions. (2 x 20 = 40 marks)

19(a) With respect to the usual metric , prove that joint convergence of a

sequence is equivalent to the marginal convergence of the

components of that sequence. (10)

19(b) State and prove Cauchy – Schwartz inequality regarding inner

product space. (10)

20(a) Let V , W be the normed vector spaces. Let f : V → W be a linear

transformation. Then prove that the following three statements are

equivalent : (16)

f is continuous on V.
There exists a point x_o in V at which f is continuous.
║f(x)║ ∕ ║x║ is bounded for x є V – { ө }.

What do we conclude from the equivalence of statements (i) & (ii)?

20(b) Show that a compact set in a metric space is complete. (4)

21(a) Prove that the number Λ is the upper limit of the sequence

{x_n , n ≥ 1 } iff for all є > 0

x_n< Λ + є for all sufficiently large n and
x_n> Λ – є for infinitely many n. (10)

21(b) Let {f_n} be a sequence of real functions integrable over the

finite interval [a , b]. If f _n→ f uniformly on [a , b],

prove that f is integrable over [a , b]. (10)

22(a) State and prove the Cauchy’s general principle of uniform

convergence of a sequence of real or complex valued functions. (8)

22(b) State and prove a necessary and sufficient condition for a function

f(x) to be Riemann – Stieltjes integrable on [a , b]. (12)

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