MAS3702 2017-2018 Example Sheet 4 PDF

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Please do the following problems:Qu, Qu, Qu, Qu, Qu and Qu;hand in your solutions on Monday the 27th of November by 16. The tutorial is on Monday the 20th November at 11 in ARMB.3.Z.A MAS3702 Linear Analysis (2017) Examples Sheet 4Qu. 43 Show thatW={f∈C[0,1]: f(0) = 0}is a closed linear subspace of ...

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Please do the following problems: Qu.43, Qu.44, Qu.48, Qu.49, Qu.52 and Qu.53; hand in your solutions on Monday the 27th of November by 16.00. The tutorial is on Monday the 20th November at 11.00 in ARMB.3.38.

Z.A.Lykova MAS3702 Linear Analysis (2017) Examples Sheet 4 Qu.43 Show that W = {f ∈ C[0, 1]: f (0) = 0} is a closed linear subspace of (C [0, 1], k· k∞ ), where kf k∞ = supt∈[0,1] |f (t)|. 20 marks Qu.44 Let c0 be the vector space of complex sequences which tend to zero and let cF be the vector subspace of those sequences having only finitely many terms different from zero. The norm k · k∞ is defined by kvk∞ = sup |vk | for v = (v1 , v2 , . . . , vk , . . . ). k∈N

Is cF a closed subspace in (c0 , k·k∞ )? Justify your answer. Show that cF is dense in (c0 , k·k∞ ). 15 marks Qu.45

Show that W = {f ∈ C[0, 1]: f (0) = 0} is a linear subspace of C[0, 1] which is not

closed with respect to the norm k· k2 , where kf k2 =

nR

1 0

|f (t)|2 dt

o1/2

.

Qu.46 Show that the subspace of polynomial functions is not closed in C[0, 1] with respect to the norm k· k∞ , where kf k∞ = supt∈[0,1] |f (t)|. Qu.47

Show that the subspace of polynomial functions is not closed in C[0, 1] with respect

to the norm k· k2 , where kf k2 =

nR

1 0

|f (t)|2 dt

o1/2

.

Qu.48 Let c0 and (ℓ∞ , k· k∞ ) be as in Qu.39 and Qu.42 respectively. Show that c0 is a closed linear subspace of ℓ∞ . 15 marks Qu.49 Show that a closed linear subspace of a Banach space is itself complete. Deduce from the previous question that (c0 , k· k∞ ) is a Banach space. 10 marks Qu.50

Define a mapping T : C2 → C2 by the formula "

#

"

#"

#

x 1 2 x T = . y 3 −4 y (i) Show that T is a linear operator. (ii) Consider C2 with the norm k· k2 , where k(x, y)k2 = (|x|2 + |y |2 )1/2 . Show that T is bounded. 1

2

Qu.51 Any matrix A ∈ Mm×n (C) gives a linear operator from Cn to Cm . For any vector v = (x1 , . . . , xn ) ∈ Cn , 





a11 . . . a1n x1    a . . . a  21 2n   x2   .  = Av =   ..   . .. ...   .   ..  am1 . . . amn xn

 Pn  a1j xj j=1  Pn a x   j=1 2j j    ..     . Pn j=1

amj xj

belongs to Cm . Thus consider the mapping T : Cn → Cm such that T v = Av for v ∈ Cn , where m,n . A ∈ Mm×n (C), A = (aij )i,j=1 (i) Show that T is a linear operator. (ii) Show that T : (Cn , k · k2 ) → (Cm , k · k2 ): v 7→ Av is bounded. Qu.52 Define a mapping T : C[0, 1] → C [0, 1] by the formula (T x)(t) = 3t2 x(t), x ∈ C [0, 1], t ∈ [0, 1]. (i) Show that T is a linear operator. (ii) Consider C[0, 1] with the norm k· k∞ , where kf k∞ = supt∈[0,1] |f (t)|. Show that T is bounded. Find the operator norm of T . 20 marks 2 2 Qu.53 Define a mapping T : ℓ → ℓ by the formula T (x1 , x2 , . . . , xn , . . . ) = (x2 , x3 , . . . , xn , . . . ). (This is called the backward shift operator or the left shift operator.) (i) Show that T is a linear operator. (ii) Show that T is bounded. Find the operator norm of T . (iii) Give an example x ∈ ℓ2 such that kxk2 = 1 and kT xk2 = 0. Qu.54

Let ℓ

∞

be as in Qu.41. Define a mapping S: cF → ℓ S(x1 , x2 , . . . , xn , . . . ) = (x1 , x1 + x2 , . . . ,

∞

n X

by the formula xk , . . . ).

k=1

(i) Show that S is well defined. Show that S is a linear operator. (ii) Consider cF with the norm k· k1 , where kxk1 =

∞ X

|xj | for x = (x1 , . . . , xn , . . . ).

j=1

Show that S is bounded. Find the operator norm of S . Qu.55 Let ℓ∞ be as in Qu.42. Define a mapping S: ℓ∞ → ℓ∞ by the formula n 1 X x1 + x2 xk , . . . ) ,... , S(x1 , x2 , . . . , xn , . . . ) = (x1 , 2 n k=1 (This is called the Cesaro operator.) (i) Show that S is well defined. Show that S is a linear operator. (ii) Show that S is bounded. Find the operator norm of S .

20 marks

3

Qu.56 Let E, F and G be normed spaces and let A: E → F and B: F → G be bounded linear operators. Consider the composition BA of operators A and B, that is, BA: E → G where (BA)(x) = B(Ax) for all x ∈ E. Show that BA is a bounded linear operator.

Qu.57 Let (V, k· k) be a normed space over C and let F : V → C be a linear functional. Show that F is continuous if and only if F is bounded....