MATH0051: Analysis 4 – Real Analysis 2019-2020
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MATH0051: Analysis 4 – Real Analysis
2019-2020
1. Let (V, | · |) be a normed space and define
d(x, y) = 
if x 
y,
if x = y.
(i) Prove that (V, d) is a metric space. (ii) Describe all convergent sequences in (V, d). (iii) Prove that (V, d) is complete.
(iv) In the case V = _ with | · | = | · |, describe the open balls Bn (0, 1), Bn (1, 1) and Bn (2, 3) in the metric space (V, d).
(v) In the case V = _ with || · || = | · |, describe all continuous functions from (V, d) to (_, | · |) and from (V, d) to (V, d).
2. Consider the series
o (ez + nl )oR2
/2o
Which of the following statements are true and which are false?
(i) The series converges pointwise for x ∈ [0, ∞).
(ii) The series converges uniformly for x ∈ [0, ∞).
(iii) The series converges pointwise for x ∈ [0, 1].
(iv) The series converges uniformly for x ∈ [0, 1].
(v) The series can be uniformly approximated for x ∈ [0, ∞) by a sequence of poly- nomials.
(vi) The series can be uniformly approximated for x ∈ [0, 1] by a sequence of poly-
nomials.
Justify your answers.
3. Which of the following subsets of C[0, 1] are compact?
(i) {cos(ax) + b : a ≥ 0, b ∈ [2, 3]} (ii) {cos(ax) + b : a ∈ [1, 2], b ≥ 0}
(iii) {cos(ax) + b : a ∈ [1, 2], b ∈ (2, 3]}
(iv) {cos(ax) + b : a ∈ [1, 2], b ∈ [2, 3]}
Justify your answers.
4. Consider the metric space (_, d), where
d(x, y) = 
Which of the following mappings T : _ → _ are contraction mappings on (_, d)?
(i) T (x) = 
;
(ii) T (x) = 
cos(x).
(iii) T (x) = sin(x);
Let T be an injective contraction mapping on (_, d).
(vi) Find the fixed point of T.
Now let ϕ : [0, ∞) → [0, ∞) be an injective contraction mapping on ([0, ∞), | · |) with a fixed point a, and consider the mapping T (x) = ϕ(|x|).
(v) For which values of a is T a contraction mapping on (_, d)?
Justify all your answers.
2022-04-27