Description

This assignment has a total of (15Marks).
Exercise 1.1
Let
U = {x ∈ R4: x1 + x2 + x3 = 0, x1 + 3×2 = x4}, V = {x ∈ R4: x1 = x4}, U + V := {x ∈ R4: x = u + v, u ∈ U, v ∈ V }.
Find dimU, dimV and dimU + V .
(3Marks)
Exercise 1.2
Calculate the pointwise limit, if it exists, for each of the following function sequences {fn}n∈N on the given domain and decide whether the convergence is uniform.
i) fn(x) = √n x, domfn = [0,1], (2Marks)
ii) , domfn = [0,∞),
(2Marks)
iii) √x, domfn = (0,∞),
Exercise 1.3
For p ∈ N {0} we define the ℓp-spaces of real sequences by

i) Prove that ℓp ⊂ ℓq for p < q. (2Marks)
ii) Find a sequence (an) such that (an) ∈ ℓp for all p > 1 but (an) ∈/ ℓ . (2Marks)
iii) Find1 a sequence (an) of real numbers such that lim an = 0 but (an) ∈/ ℓp for all p ∈ [1,∞). n→∞ (2Marks)