MA244 Analysis III
Assignment 4.
15% of the credit for this module will come from your work on four assignments submitted
by a 3pm deadline on the Monday in weeks 4,6,8,10. Each assignment will be marked out
of 25 for answers to two randomly chosen ’B’ and two ’A’ questions. Working through
all questions is vital for understanding lecture material and success at the exam. ’A’
questions will constitute a base for the first exam problem worth 40% of the final mark,
the rest of the problems will be based on ’B’ and ’C’ questions.
The answers to ALL questions are to be submitted by the deadline of 3pm on
Monday, the 1st of December 2014. Your work should be stapled together,
and you should state legibly at the top your name, your department and the
name of your supervisor or your teaching assistant. Your work should be deposited in your supervisor’s slot in the pigeon loft if you are a Maths student,
or in the drop box labeled with your teaching assistant’s name, opposite the
Maths Undergraduate Office, if you are a non-Maths or a visiting student.
0.1
Norms
1. A. In our course we only study norms on real vector spaces. Let V be a complex
vector space. A function || · || : V → R is called a norm if:
• ||z|| ≥ 0 for all z ∈ V
• If ||z|| = 0, then z = 0
• If λ ∈ C and z ∈ V , then ||λz|| = |λ| · ||z|| (Here |λ| is the modulus of λ ∈ C)
• If z, w ∈ V , then ||z + w|| ≤ ||z|| + ||w||.
Which of the following functions are norms on Cn ?
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(a) N1 (z) = nk=1 (zk + z̄k );
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(b) N2 (z) = ( nk=1 z̄k zk )1/2 .
(Here z = {zk = xk + iyk }nk=1 , z̄ = {xk − iyk }nk=1 ∈ Cn ). Justify your answers
2. A. A pair (X, d) is called a metric space if X is a set and d : X × X → R is a
function with the following properties: for any x, y, z ∈ X,
• d(x, y) ≥ 0 (non-negativity)
• d(x, y) = 0 iff x = y (separation of points)
• d(x, y) = d(y, x) (symmetry)
• d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)
Let (V, || · ||) be a normed vector space. Define d(x, y) := ||x − y||, x, y ∈ V . Prove
that (V, d) is a metric space.
3. A. Let aj ∈ R for 1 ≤ j ≤ n and write
||x||a =
n
X
j=1
aj |xj |.
State and prove necessary and sufficient conditions for || · ||a to be a norm on Rn .
4. B. Consider SF , the space of real sequences a = (an )∞
n=1 , such that all but finitely
many of the an ’s are zero. (In other words, each sequence a ∈ SF is eventually
zero.)
(a) Show that if we use the natural definition of addition and scalar multiplication
(an ) + (bn ) = (an + bn ), λ(an ) = (λan ), λ ∈ R,
then SF is a vector space.
(b) Show that the following definitions all give norms on SF ,
||a||∞ = max |an |,
(1)
||a||w = max |nan |,
(2)
||a||1 =
(3)
n≥1
n≥1
∞
X
|an |,
n=1
||a||2 =
||a||u =
∞
X
!1/2
|an |2
,
n=1
∞
X
n|an |.
(4)
(5)
n=1
(c) Show that norms (2), (3), (4), (5) are NOT equivalent to norm (1). (In
fact no two of norms from the above list are equivalent. Verify this if you
accidentally got a free Saturday night. At least, check the most interesting
case of (2) and (3).)
0.2
Completeness.
5. A. Let SF be the space defined in Question 4. Prove that the normed vector space
(SF , || · ||1 ) is not Banach. Here || · ||1 is the norm defined in (3).
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6. B. Let l1 be the set of real sequences a with ∞
j=1 |aj | convergent.
(a) Show that l1 is a vector space given the natural definitions of addition and
multiplications:
(an ) + (bn ) = (an + bn ), λ(an ) = (λan ), λ ∈ R.
(b) Prove that (l1 , || · ||1 ) is a complete normed vector space (Banach space). Here
|| · ||1 is the function defined in (3).You may assume without a proof that || · ||1
is a norm on l1 .
7. A. A subset X of a normed vector space (V, || · ||) is said to possess the BolzanoWeierstrass property if every sequence (xn )∞
n=1 ⊂ X has a convergent subsequence
k→∞
xnk −→ x ∈ X. Prove that X is complete in the sense that every Cauchy sequence
in X converges to a point in X.
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8. A. This question is a preparation for the discussion of the Banach contraction
mapping theorem. Let (X, || · ||) be a normed vector space and T : X → X a
mapping. We say that w ∈ X is a fixed point of T if T w = w. We say that T is
a contraction mapping if there exists a positive number K < 1 with ||T x − T y|| ≤
K||x − y|| for all x, y ∈ X. Suppose that x0 ∈ X and define xn inductively by
xn = T xn−1 , n > 0. Show that the sequence (xn )n≥1 converges to w for any
x0 ∈ X.
0.3
Closed and open sets, continuity
9. B. Consider the real normed vector space l∞ of bounded real sequences a = (a1 , a2 , . . .)
with norm ||a||∞ = supn≥1 |an |. The linear structure is given by component-wise
addition and scalar multiplication defined in Question 4(a).
(a) Show that the set
E = {a ∈ l∞ : ∃N (a) : ∀n > N (a), an = 0}.
is a linear subspace of l∞ , but not a closed subset.
(b) Show that the set
F = {a ∈ l∞ : a2n = 0 ∀n = 1, 2, 3, . . .}.
is a closed infinite-dimensional linear subspace of l∞ .
(c) Show that any finite-dimensional linear subspace of any normed vector space
is closed.
10. B. Consider the vector space SF defined in Question 4.
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(a) Show that the map T : SF → R defined by T a = ∞
j=1 aj is linear.
(b) B. If we equip R with the usual Euclidean norm | · | and SF with one of the
five norms (1)-(5), state, with reasons, whether T is continuous.
11. C. Let UP
= V = SF , where SF a normed space defined in Question 4. Let ||a||V =
||a||U = ∞
n=1 |an |.
(a) Show that if T : U → V is defined by T a = b with bj = (1 − j −1 )aj , j =
1, 2, 3, . . ., then T is a continuous linear map. However, prove that there does
not exist an a ∈ U with a 6= 0 such that ||T a||V = ||T ||||a||U .
(b) If U and V are two finite-dimensional normed vector spaces and T : U → V is
linear, can we always find an a ∈ U with a 6= 0 such that ||T a||V = ||T ||||a||U ?
Give reasons.
17th of November 2014
Daniel Ueltschi and Oleg Zaboronski
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