MAS3702 2017-2018 Example Sheet 3 PDF

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Please solve the following problems:Qu, Qu(i), Qu(ii), Qu(v) and Qu;hand in your solutions on Monday the 13th of November by 16. The tutorial is on Monday the 6th November at 11 in ARMB.3.Z.A MAS3702 Linear Analysis (2017) Examples Sheet 3Qu. 33 Show thatxn=n/(n+ 1) is a Cauchy sequence in (R,|·|).Q...

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Please solve the following problems: Qu.35, Qu.36(i), Qu.36(ii), Qu.36(v) and Qu.39; hand in your solutions on Monday the 13th of November by 16.00. The tutorial is on Monday the 6th November at 11.00 in ARMB.3.38.

Z.A.Lykova MAS3702 Linear Analysis (2017) Examples Sheet 3 Qu.33

Show that xn = n/(n + 1) is a Cauchy sequence in (R, | · |).

Qu.34 Give an example of a Cauchy sequence in (R, | · |) such that xn ∈ (0, 1) for all n ≥ 1 which does not converge in (0, 1). Qu.35

Show that the sequence vn = (3n /(3n + 5n ), 3n /(3n + 2)) tends to v = (0, 1) as

n → ∞ in (C2 , k· k2 ) where, for v = (v1 , v2 ) , kvk2 = Qu.36

n

|v1 |2 + |v2 |2

o1/2

. 12 marks

Consider the complex vector space C[0, 1] and the following norms on C[0, 1]: kf k∞ = sup |f (t)|; kf k1 = t∈[0,1]

Z 1 0

|f (t)| dt and kf k2 =

Z 1 0

|f (t)|2 dt

1/2

.

(i) Show that the sequence of functions fn (t) = e−nt converges to g(t) = 0 as n → ∞ with respect to the norm k· k2 . 16 marks −nt (ii) Show that fn (t) = e does not tend to this limit with respect to the norm k· k∞ . 12 marks n (iii) Show that fn (x) = x is a Cauchy sequence with respect to the norm k· k2 . Does this sequence tend to a limit in (C[0, 1], k· k2 )? (iv) Show that fn (x) = xn is a Cauchy sequence in (C[0, 1], k· k1 ). Does this sequence tend to a limit? (v) Show fn (x) = xn does not tend to any limit in (C[0, 1], k· k∞ ). 20 marks Qu.37 Assume that xk → x and xk → y as k → ∞ in the normed space (X, k· k). Show that for every ǫ > 0 there exists an N0 such that for all k ≥ N0 we have kxk − xk < ǫ and 



Hence show that x = y. [Hint: consider x − y  .]

Qu.38 space.

  xk



− y  < ǫ.

Show that Cn , the complex vector space of n-tuples of complex numbers, is a Hilbert

Qu.39 Show that c0 , the complex vector space of complex sequences which tend to zero with the norm k · k∞ , is complete, that is, is a Banach space, where kvk∞ = max |vi | for v = (v1 , . . . , vn , . . . ). i∈N

40 marks Qu.40 space.

Show that the normed space (C[0, 1], k· k1 ) is not complete, and so is not a Banach 1

Qu.41 Show that the normed space ℓ1 , the complex vector space of complex sequences P∞ ∞ |xj | < ∞ with the norm k · k1 , is complete, that is, is a Banach x = (xn )n=1 such that j=1 space, where kxk1 =

∞ X

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

j=1

Qu.42 Prove that the normed space (ℓ∞ , k · k∞ ), the complex vector space of all bounded sequences of complex numbers, is a Banach space which is not a Hilbert space, where kxk∞ = sup |xn | for x = (x1 , . . . , xn , . . . ). n∈N...