LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – NOVEMBER 2003
ST-1800/S715 – ANALYSIS
04.11.2003 Max:100 marks
1.00 – 4.00
SECTION-A
Answer ALL questions. (10×2=20marks)
- Let Z be the set of all integers. Construct a function form Z to Z which is not one to one and also not onto.
- Define a metric on a non-empty set x.
- The real valued function f on R2 – is defined by f (x, y) = . Show that
lim f (x,y) does not exist as (x, y) (0, 0) .
- State weirstrass’s approximation theorem.
- If is a convergent sequence in a metric space (X, P) then prove that it is a cauchy sequence.
- If Un = O (1/nk-2), for what value of k converges?
- Define the upper limit and lower limit of a sequence.
- Find and also the double limit of xmn as m,n where xmn = .
- Let f: Rm. Define the linear derivative of f at .
- From the infinite series where obtain the expansion for log (1+x).
SECTION-B
Answer any FIVE questions. (5×8=40marks)
- Show that the space R’ is complete.
- State and prove cauchy’s inequality.
- Prove that the union of any collection of open sets is open and the intersection of any collection of closed sets is closed.
- a) Show that fn (x) = x is not uniformly convergent
- b) Let and be metric spaces. Let the sequence fn : converge to f uniformly on x. If C is a point at which each fn is continuous, then show that f is continuous at C.
- Let V, W be normed vector spaces. If the function f : V W is linear, then show that the following statements are equivalent.
- f is continuous on V
- there is a point at which f is continuous.
- is bounded for
- Examine for convergence of if
- un =
- Let (be a metric space and let f1, f2, …..fn be functions on X to R‘. The function
f = (f1, f2, …..fn) : is given by f(x) = (f1(x) … fn (x). Prove that f is continuous at x0 if and only if f1, f2,…..fn are continuous.
- If f : is differentiable at then prove that the linear derivative of f at is unique.
SECTION-C
Answer any TWO questions. (2×20=40marks)
- a) Let (and be metric spaces. Prove that the following condition is
necessary and sufficient for the function f : to be continuous on X:
whenever G is open in Y, then f-1 (G) is open in X.
- b) Show that if is a metric on x then so is given by (x, y) = and P and
are equivalent. (12+8)
- a) State and prove Banach’s fixed point theorem.
- b) State and prove Heine – Borel theorem. (10+10)
- a) State and prove d’ alembert’s ratio test
- b) Discuss the convergence of where
- c) Discuss the convergence and absolute convergence of
(8+8+4)
- a) Show that a necessary and sufficient condition that fis that, given
there is a dissection D of [a, b] such that S (D, f, g) – s (D, f, g) < .
- b) If fI, f2 R [g i a, b] then prove that f1 f2 R [g i a, b]
- c) If f R [ g i a, b] then show that (7+7+6)
Latest Govt Job & Exam Updates: