1. hw 7
(1) Assuming that the Bolzano-Weierstrass Theorem is true for R1, prove that the Bolzano- Weierstrass Theorem is true for Rp for any p ∈ N by mathematical Induction. (Hint: Let (an) be a bounded sequence in Rp+1. Write an = (xn, yn) for all n ≥ 1 where (xn) is a sequence in Rp and (yn) is a sequence in R. Show that (xn) is bounded in Rp and (yn) is bounded in R.)
(2) Let (M, d) be a complete metric space and N be a subset of M. Let (N, dN) be the metric subspace of (M, d) associated with N. Prove that (N, dN) is complete if and only if N is a closed subset of M.
(3) In Calculus, we have learned the following important properties for convergent sequences:
Theorem 1.1. Let (an) and (bn) be sequence of real numbers and k be a real number.
Suppose that
n→∞lim an= a, lim
n→∞bn = b.
Then (a) lim
n→∞(an+ bn) = a + b;
(b) lim
n→∞(kan) = ka;
(c) lim
n→∞anbn= ab (d) lim
n→∞
an bn
= a
b if b ≥ 0.
In class, we also proved
Theorem 1.2. Let (an) be a sequence in Rp with an = (a1n, · · · , apn) for n ≥ 1. Then (an) is convergent in Rp if and only if (a1n), · · · , (apn) are convergent in R. In this case,
n→∞lim an=
n→∞lim a1n, · · · , lim
n→∞apn .
Let (xn) and (yn) be sequences in Rpsuch that limn→∞xn= x andlimn→∞yn= y. Use Theorem 1.1 and Theorem 1.2 to prove the following statements:
(a) lim
n→∞(xn+ yn) = x + y.
(b) lim
n→∞kxn= kx.
(c) lim
n→∞anxn = ax.
(d) lim
n→∞
xn bn
= x
b if b ≥ 0.
(e) lim
n→∞hxn, yni = hx, yi.
(4) Do project in Bartle’s Book. 8β (a), (b), (c) on page 61.
(5) Let 1 ≤ p < ∞ and lp(N) be the space of sequences a : N → R of real numbers such that P∞
n=1|a(n)|p< ∞. Define a function k · k : lp(N) → R by kak =
∞
X
n=1
|a(n)|p
!1/p
.
(a) Prove that lp(N) is a vector space over R. (Prove that lp(N) is a vector subspace of the space of real valued functions F (N, R).)
(b) Prove that k · k is a norm on lp(N). This norm is denoted by k ·klp(N). (You need exercise (4)).
(c) Let us denote the norm space (lp(N), k · klp(N)) simply by lp(N). Prove that lp(N) is a Banach space over R.
(d) Let α be a real number and a : N → R be the function a(n) = 1/nα. Determine all α so that a ∈ lp(N).
(e) Prove or disprove that lp(N) ⊂ lq(N) for p > q.
(f) Prove that lp(N) ⊂ l∞(N) for all 1 ≤ p < ∞.
1
2
(6) Let a and b be two vectors in l2(N).
(a) Show that P∞
n=1a(n)b(n) is absolutely convergent in R. (Hint: use Cauchy-Schwarz inequality; see (4)).
(b) Define
ha, bi =
∞
X
n=1
a(n)b(n).
Prove that h·, ·i defines an inner product on l2(N) so that ha, ai = kakl2(N). By Exercise (5) and this exercise, l2(N) is a Hilbert space over R.
(c) Two vectors v, w in an inner product space V is orthogonal if hv, wi = 0. For each i ∈ N, define ei: N → R by (ei)(n) = δin for all n ∈ N. Here δij is the number defined by
δij =
(1 if i = j 0 otherwise.
Prove that {ei : i ∈ N} is an orthonormal family of vectors in l2(N) (in other words, elements of {ei} are unit vectors and are orthogonal to each other) and compute the distance kei− ejkl2(N)for any i 6= j.
(d) Let S∞= {a ∈ l2(N) : kak = 1}. Prove that S is closed and bounded.
(e) Show that (ei) is a sequence in S∞ and does not have any convergent subsequence in S∞. (This example tells us that in a metric space, a closed and bounded subset need not be sequentially compact).
(f) Let fi= ei−2ei+1for all i ≥ 1. Determine the orthogonal complement V of {fi : i ≥ 1}, i.e.
V = {a ∈ l2(N) : ha, fii = 0, for all i ≥ 1}.
(g) Let gi = ei − 3ei+1 for all i ≥ 1. Determine the orthogonal complement of W of {gi: i ≥ 1}, i.e.
W = {a ∈ l2(N) : ha, gii = 0, for all i ≥ 1}.
(h) Let hi = ei− 5ei+1+ 6ei+2 for all i ≥ 1. Determine the orthogonal complement Z of {hi: i ≥ 1}, i.e.
Z = {a ∈ l2(N) : ha, hii = 0, for all i ≥ 1}.
Prove or disprove that Z is the direct sum of V and W. In other words, is it true that V ∩ W = {0} and Z = V + W ?
(i) Let a be a vector in l2(N). For each n ≥ 1, we define sn= a(1)e1+a(2)e2+· · ·+a(n)en. Then (sn) is a sequence of vectors in l2(N). Show that (sn) is convergent to a in l2(N).
In this case, we denote
(1.1) a =
∞
X
n=1
a(n)en.
The expression 1.1 for a is called the Fourier expansion of a with respect to the or- thonormal family {ei: i ∈ N}.
(j) Let (ak) be a sequence of vectors in l2(N). By (6i), we can write ak =
∞
X
n=1
ak(n)en. Prove or disprove that lim
k→∞ak = a in l2(N) if and only if lim
k→∞ak(n) = a(n) for any n ≥ 1. (Compare to (3)).
(7) Let R∞ be the subset of F (N, R) consisting of sequences a : N → R such that there exists N ∈ N so that a(n) = 0 for all n ≥ N.
(a) Prove that R∞ is a vector subspace of lp(N) for all 1 ≤ p ≤ ∞.
(b) Prove that R∞ is not a closed subset (subspace) of lp(N) for all 1 ≤ p ≤ ∞. (This implies that (R∞, k · kp) is not a Banach space, where kakp = kaklp(N) for a ∈ R∞ by exercise (2).)
