Lecture Note:

Math 2111 Advanced Calculus (I)

Homework 8 Hand in Problems:

1, 2, 3(a)(e), 4(a), 5, 6, 8 Lecture Note: 1, 2, 4 1. Let {a_n}^∞_n=1 be a sequence in R with an> 0 and 0 < lim inf

n→∞ a_n≤ lim sup

n→∞

a_n < ∞. Prove that

lim sup

n→∞

1

a_n = 1

lim inf_n→∞a_n.

2. Let {a_n}^∞_n=1 be a bounded sequence in R and f : R → R be a continuous and increasing function. Prove that

lim sup

n→∞

f (a_n) = f lim sup

n→∞

a_n
.

3. Check that the following spaces with the given functions k · k are normed vector spaces.

Let I ⊆ R be an interval and a = (a1, a₂, a₃, · · · ) with a_i ∈ R for i = 1, 2, · · · ..

(a) L²(I) :=n

f : I → R   

R

I|f (x)|²dx < ∞o

with kf k₂ := R

I|f (x)|²dx1/2

. (b) L^∞(I) :=n

f : I → R    sup

x∈I

|f (x)| < ∞o

with kf k∞:= sup

x∈I

|f (x)|.

(c) C(I) :=n

f : I → R  

 f is continuous on I and sup

x∈I

|f (x)| < ∞o

with kf kC(I):= sup

x∈I

|f (x)|.

(d) `² :=

n a

∞

X

i=1

|ai|² < ∞ o

with kak_`² :=

X^∞

i=1

|ai|²1/2

.

(e) `^∞ :=n a

   sup

i∈N

|a_i| < ∞o

with kak_`^∞ := sup

i∈N

|a_i|.

4. Check that the following spaces with the given functions < ·, · > are inner product spaces.

Let I ⊆ R be an interval and a = (a¹, a2, a3, · · · ) with ai ∈ R for i = 1, 2, · · · . (a) L²(I) with < f, g >:=

Z

I

f (x)g(x) dx.

(b) `² with < a, b >:=

∞

X

i=1

a_ib_i.

5. Let (V, k · k) be a normed vector space and {v_k}^∞_k=1 is a sequence in V (a) Show that if lim

k→∞v_k = v, then

lim

k→∞kv_kk = kvk.

(b) Give an example to show that the converse of (a) is false.

6. Let {r_n}^∞_n=1 be a sequence in R³ where

r_n =







(a, 0, 0) if n = 3k − 2,

(0, b, 0) if n = 3k − 1, k ∈ N (0, 0, c) if n = 3k

Prove that

∞

X

n=1

r_n

n! converges in R³with the usual norm kxk =X³

i=1

x²_i1/2

where x = (x₁, x₂, x₃).

(Note: if using the Cauchy criterion, you may need to check that (R³, k · k) is a Banach space.)

7. Suppose that we know the fact that

L² [0, 1]
, k·k₂ ,

C [0, 1]
, k·kC([0,1])

 and

L^∞ [0, 1]
, k · k∞

 are Banach spaces. Let f_k(x) = x^k for k = 0, 1, 2, · · · . Use the Cauchy criterion to prove

that (a)

∞

X

k=0

f_k(x)

2^k converges in 

L² [0, 1]
, k · k₂ .

(b)

∞

X

k=0

fk(x)

k! converges in 

C [0, 1]
, k · kC([0,1])

 .

(c)

∞

X

k=1

f_k(x)

k² converges in 

L^∞ [0, 1]
, k · k∞

 .

8. Let (V, k · k) be a normed vector space and W ⊆ V .

(a) Prove that if W is closed in V , then for any convergent sequence {v_k}^∞_k=1 ⊆ W , the limit v = lim

k→∞vk also belongs to W .

(b) Let V = C [0,¹₂]
with the norm kf k := max

x∈[0,¹₂]

|f (x)|. Prove that

∞

X

n=0

1 n + 1xⁿ converges in 

V, k · k

. (Note: Suppose that 

V, k · k

is a Banach space.)

(c) Let W be the set consisting of all polynomials on [0,¹₂] and P_k(x) =

k

X

n=0

1 n + 1xⁿ. Use (b) to prove that W is not closed in C [0,¹₂]
.

(d) Given δ > 0, define

g(x) := δ

∞

X

n=1

1

n + 1xⁿ, x ∈ [0,1 2].

Prove that g ∈ B(0, δ) where 0 is the zero function on [0,¹₂].

(e) Use (d) to prove that the set W in (c) is not open in C [0,¹₂]
.

9. Let {a_n}^∞_n=1 be a sequence in R.

(a) Prove that

lim sup

n→∞

a_n= max

lim sup

n→∞

a_2n, lim sup

n→∞

a_2n+1 .

(b) Suppose that lim sup

n→∞

a2n+1

a_2n < 1 and lim sup

n→∞

a2n+2

a_2n+1 < 1. Determine whether the series

∞

X

n=1

a_n converges.

(c) Suppose that lim sup

n→∞

a_2n+1 a_2n

< 1 and lim sup

n→∞

a_2n+2 a_2n+1  

< 1. Determine whether the series

∞

X

n=1

a_n converges.

Lecture Note:

• (Page 50)

1. Problem 1.22

• (Page 98)

2. Problem 2.15 3. Problem 2.16

4. Problem 2.17(1)(3)(5)