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DEPARTMENT OF MATHEMATICAL SCIENCES, IIT ( BHU ) EVEN SEMESTER 2020-21 MA313 – REAL AND COMPLEX ANALYSIS Instructions: Use a mathematical word-processor (e

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DEPARTMENT OF MATHEMATICAL SCIENCES, IIT ( BHU )

EVEN SEMESTER 2020-21

MA313 – REAL AND COMPLEX ANALYSIS

Instructions:

• Use a mathematical word-processor (e.g. TeX) or write on A4 paper, take careful photographs and assemble them into a single pdf file (there are many apps available for doing this).
• Start each question on a new page. Write your roll number in the top right corner of each page. Use only one side of each page(for clear photo).
• Write your name, department and roll number clearly at the top of the first page.
• Answer all questions. Give proper justification. Don’t copy from others.

1. Let F be a finite field. Is there an order on F which turns it into an ordered field?

Justify. (2 Marks)

2. Let X = [0,1]∪(2,3) with the metric d(x,y) = |x − y|. Let A = [0,1] and B = (2,3). Then A,B ⊆ X. In the metric space (X,d):
   1. Is A open? Justify.            (1 Mark)
   2. Is A compact? Justify.    (1 Mark)
   3. Is B closed? Justify.         (1 Mark)
   4. Is B bounded? Justify.    (1 Mark)
   5. Is B compact? Justify.    (1 Mark)
3. Let q ∈ N,n ∈ Z. Let |n|_q be defined as in the example at the end of Lecture 9 (27/01/2021). Let p be a prime number.
   1. Prove that |mn|_p = |m|_p|n|_p. Give an example of m,n ∈Z such that |mn|₄ 6=

|m|₄|n|₄.           (2 Marks)

 

, let

. Prove that |r|_p is a well defined, i.e. that if

 

 

 then

.         (1 Mark)

 

1. 3. Prove that |r + s|_p ≤|r|_p + |s|_p for all r,s ∈Q.       (1 Mark)
   4. If r,s ∈Q, define d_p(r,s) = |r − s|_p. Prove that (Q,d_p) is a metric space.

(1 Mark)

5. In the metric space (Q,d₃), consider the sequence

 

.

 

Is {x_n} Cauchy? Does {x_n} converge? If so, find the limit.    (3 Marks)

6. Is Z a closed set in the metric space (Q,d₃)? Justify your answer. (1 Mark)

4. Let X be a non-empty set, and suppose d and d⁰ are metrics on X. We say that d and d⁰ are equivalent if there exist c₁,c₂ ∈ (0,∞) such that

c₁d(x,y) ≤ d⁰(x,y) ≤ c₂d(x,y),

for all x,y ∈ X.

Assume that d and d⁰ are equivalent.

4. 1. Let {p_n} be a sequence in X. Prove that {p_n} converges in (X,d) iff {p_n} converges in (X,d⁰).           (1 Mark)
   2. Prove that a set E is open in (X,d) iff E is open in (X,d⁰).    (2 Marks)
   3. Prove that a set K is compact in (X,d) iff K is compact in (X,d⁰). (2 Marks)
5. Let (X,d_X) and (Y,d_Y ) be metric spaces. Let Z = X × Y , the cartesian product of the sets X and Y . Define d₂ : Z × Z →R and d_∞ : Z × Z →R by

 

         and

 

.

1. 1. Prove that d₂ and d_∞ are metrics on Z.              (2 Marks)
   2. Prove that d₂ and d_∞ are equivalent metrics. (2 Marks)