MATH0051: Analysis 4 – Real Analysis
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH0051: Analysis 4 – Real Analysis
1. (a) Let
fn (x) = , 1(n)2x2
if
if
0 ≤ x ≤ ,
< x ≤ 1.
Does the sequence (fn )n41(o) converge pointwise on [0, 1]?
If it does, what is the limit function?
Does the sequence (fn )n41(o) converge uniformly on [0, 1]?
Justify your answers.
(b) Can the series
o (cos(nx) + n3 )1/扌
扌
be approximated to arbitrary precision by a polynomial, uniformly for x ∈ [0, 1]? Justify your answer.
(c) Are either of the sets
(i) {sin(ax) + b : a > 0, b ∈ [1, 3]}
(ii) {sin(ax) + b : a ∈ [1, 2], b > 0}
compact in (C[0, 1], |.|Ul— )?
Justify your answers.
2. Let (X, d) be a metric space.
(a) Let (Kn ) be a sequence of non-empty compact subsets of X such that Kn八1 ∈ Kn for all n ∈ N. Prove that n41(o) Kn is non-empty.
(b) Let K be a non-empty compact subset of X and let x ∈ X. Show that there exists y* ∈ K such that d(x, y* ) = infyeK d(x, y).
(c) Given S ∈ X and e > 0 let Se = {x ∈ X : d(x, y) ≤ e for some y ∈ S}.
(i) Show that if e, eo > 0 then (Se )e/ ∈ Se八e/ . Is it always true that (Se )e/ = Se八e/ ? (ii) Show that if S is compact then Se is closed and bounded.
(iii) Is it always true that if S is compact then Se is compact? Justify your
answer.
(iv) Describe Se in the case where X = R2 , d is the metric induced by the norm
|.|o , and S = {(x, 0) : x ∈ [0, 1]}.
3. For non-empty subsets X and Y of R, define
, ←
Let K denote the set of all non-empty compact subsets of (R, I.I).
(a) Show that (K, d) is a metric space.
Hint : Note that by Q2(b) when X, Y ∈ K the infima in the definition of d(X, Y) can be replaced by minima (i.e. the infima are attained).
(b) Show that, for X, Y ∈ K and e > 0,
d(X, Y) ≤ e ← X ∈ Ye and Y ∈ Xe ,
where, for a non-empty subset X of R and e > 0,
Xe = {y ∈ R : Ix - yI ≤ e for some x ∈ X}.
(c) Which of the following sets is contained in the closed ball B([0, 1], 1/4)?
(i) [-1/4, 3/4] (ii) {1/2}
Justify your answers.
(d) Show that (K, d) is complete.
Hint: Given a Cauchy sequence (Xn ) in (K, d), define
o
X = Xn ,
N41 n/N
then show that X ∈ K and d(Xn , X) → 0 as n → &.
4. Let T : R → R be defined by
T (x) = sin x.
Let (K, d) be as defined in Q3.
(a) Show that T is a contraction mapping on (R, I.I), and hence also on (K, d), when
we define T (S) = {T (x) : x ∈ S} ∈ R for S ∈ R.
(b) Define
厂(S) = T (S) U (1/2 + T (S)) for S ∈ R,
where 1/2 + T (S) = {1/2 + T (x) : x ∈ S} ∈ R.
Use part (a) to show there exists a unique K ∈ K such that 厂(K) = K .
Hint: First show that d(X1 U X2 , Y1 U Y2 ) ≤ max (d(X1 , Y1 ), d(X2 , Y2 )) for X1 , X2 , Y1 , Y2 ∈ K.
(c) Show that if Ko ∈ K satisfies 厂(Ko ) ∈ Ko then the set K from part (b) satisfies
o
K = 厂k (Ko ),
k4o
where 厂o (Ko ) = Ko and 厂k (Ko ) = 厂(厂k -1 (Ko )) for k ∈ N.
2022-04-27