MT 1816 – REAL ANALYSIS

(12 BATCH STUDENTS ONLY)

Date : 05/11/2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Answer all the questions. Each question carries 20 marks.

I.a) 1) Prove that refinement over an interval increases the lower sum and decreases the upper sum.

a)2. Using the notion of Upper sums and Lower sums of a bounded function, when do you say that on [a,b]. When does the Riemann-Stieltjes integral reduces to Riemann integral? (5)

b)1) Suppose f is bounded on [a,b]. f has finitely many points of discontinuity on [a,b] and is continuous at every point at which f is discontinuous the prove that .

b)2) Prove: Let be a monotonically increasing function on [a,b] and let on [a,b]. If f is a bounded real function on [a,b] then prove that on [a,b] on [a,b]. In this case (5 +10)

c)1) Prove: on [a,b]

c)2) Suppose c Î (a,b) and two of the three integrals , and exist. Then prove that the third also exists and

c)3) If f is monotonic on [a,b] and if is continuous on [a,b], then prove that . (4+4+7)

a)1) State and prove the theorem on Cauchy criterion for uniform convergence.

a)2) Using a suitable example show that the limit of the integral need not be equal to the integral of the limit. (5)

b)1) If X is a metric space and denote the set of all complex valued, continuous, bounded functions with domain X. Choosing a suitable distance function between the elements of prove that is a metric space and also a complete metric space.

b)2) Suppose that is a sequence of differentiable functions on [a,b]. Suppose that converges at some point . If converges uniformly on [a,b] then prove that converges uniformly to some function f and (5+10)

c)1) Let . Verify whether .

c)2) Suppose converges uniformly to a function f on E, where E is a set as a metric space. Let x be a limit point of E and suppose that . Then prove that {A_n}conveges and (5+10)

III. a)1) State and prove Parseval’s formula.

a)2) State and prove Dini’s theorem. (5)

b)1) State and prove Jordan’s theorem.

b)2)State and prove Riemann- Lebesgue Lemma. (6+9)

c)1) State and prove Riesz – Fischer’s theorem.

c)2) State and prove Riemann Localization theorem. (6+9)

a)1) Suppose E is an open set in Rⁿ and f maps E into R^m and x is an element in E such that when , . Then prove that A is unique.

a)2) Let be the set of all invertible linear operators on Rⁿ. If then prove that . (5)

b) State and prove Inverse function theorem. (15)

c)1) If and c is a scalarthen prove that . And with the distance between A and B defined as , prove that is a metric space.

c)2) Define Contraction principle and prove the following theorem: Let X is a complete metric space and if is a contraction of X into X then there exists only one xX such that (x)=x. (5+10)

Va)1). Derive the rectilinear co ordinates.

a)2) Derive the sum of powers of . (5)

b)1) Explain how the product and quotient rule are derived for functions f(x) and g(x)?

b)2) Derive the expression for D’ Alembert’s wave equation for a vibrating string.

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