ℓp spaces - Hölder and Minkowski inequalities - Question Bank

1. What is the ℓp norm of the sequence x = (2, 0, 0, 0, ...)?
A) 2
B) 2^p
C) 2^(1/p)
D) 1
2. Consider the sequence x = (1, 1, 1, ...). Which statement is true about its membership in ℓp?
A) x ∈ ℓp for all p ≥ 1
B) x ∈ ℓp for p > 1
C) x ∈ ℓp only for p = ∞
D) x ∈ ℓp for p = 1
3. What is the relationship between ℓp and ℓq when p < q?
A) ℓp ⊂ ℓq
B) ℓq ⊂ ℓp
C) ℓp = ℓq
D) No strict inclusion holds in general for infinite sequences
4. Which of the following is NOT a property of the ℓp norm for 1 ≤ p < ∞?
A) ||x||_p ≥ 0
B) ||x||_p = 0 if and only if x is the zero sequence
C) ||cx||_p = c ||x||_p for any scalar c
D) ||x + y||_p ≤ ||x||_p + ||y||_p
5. If x ∈ ℓp and ||x||_p = 0, what can we conclude about the sequence x?
A) x = (1, 0, 0, ...)
B) x is the zero sequence (0, 0, 0, ...)
C) x = (1, 1, 1, ...)
D) x can be any sequence
6. What is the significance of Minkowski's inequality in the context of ℓp spaces?
A) It defines the norm
B) It proves completeness
C) It establishes the triangle inequality for the norm
D) It relates the norms of sequences under term-wise multiplication
7. What is the significance of Hölder's inequality in the context of ℓp spaces?
A) It defines the norm
B) It proves completeness
C) It relates the norms of sequences under term-wise multiplication
D) It establishes separability
8. Consider the sequence x = (a, a, a, ...). For which p is x in ℓp?
A) Only if a = 0
B) For all p if the sequence is finite
C) Only if p < ∞ and the sequence is finite
D) Never if a ≠ 0 and the sequence is infinite
9. What is the value of p for which ℓp and ℓp' are the same space (up to isomorphism)?
A) p = 1
B) p = 2
C) p = ∞
D) Never
10. If x ∈ ℓp and y ∈ ℓq such that 1/p + 1/q = 1, then the dot product ∑ x_n y_n is:
A) Defined and bounded by ||x||_p ||y||_q
B) Always zero
C) Undefined
D) Bounded by ||x||_p + ||y||_q
11. The dual space of ℓ∞ is isomorphic to:
A) ℓ1
B) ℓ∞
C) ℓp'
D) ℓ2
12. The dual space of ℓ1 is isomorphic to:
A) ℓ1
B) ℓ2
C) ℓp'
D) ℓ∞
13. The dual space of ℓp for 1 < p < ∞ is isomorphic to:
A) ℓp
B) ℓ1
C) ℓp'
D) ℓ∞
14. What is the ℓ2 norm of the sequence (1, 1, 1, ...)?
A) 1
B) 2
C) ∞
D) Undefined
15. What is the ℓ1 norm of the sequence (1, 1/2, 1/4, 1/8, ...)?
A) 2
B) 1
C) 4
D) ∞
16. If p = 1, what is the Hölder inequality ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_1 ||y||_∞?
A) ∑_{n=1}^∞ |x_n y_n| ≤ (∑_{n=1}^∞ |x_n|) * (sup_{n} |y_n|)
B) ∑_{n=1}^∞ |x_n y_n| ≤ (∑_{n=1}^∞ |x_n|) + (sup_{n} |y_n|)
C) ∑_{n=1}^∞ |x_n y_n| ≤ max(∑_{n=1}^∞ |x_n|, sup_{n} |y_n|)
D) ∑_{n=1}^∞ |x_n y_n| ≤ (∑_{n=1}^∞ |x_n|)^∞ * (sup_{n} |y_n|)
17. Consider x = (1, 1/2, 1/3, ...) and y = (1, 1/2, 1/3, ...). What is ∑_{n=1}^∞ x_n y_n?
A) π^2 / 6
B) 1
C) ∞
D) ζ(2)
18. Which inequality is fundamental for proving that the sum of two elements in ℓp is also in ℓp?
A) Hölder's inequality
B) Minkowski's inequality
C) Cauchy-Schwarz inequality
D) Jensen's inequality
19. The Minkowski inequality can be written in integral form for L^p spaces as:
A) (∫|f+g|^p dx)^(1/p) ≤ (∫|f|^p dx)^(1/p) + (∫|g|^p dx)^(1/p)
B) (∫|f+g|^p dx)^(1/p) ≥ (∫|f|^p dx)^(1/p) + (∫|g|^p dx)^(1/p)
C) (∫|f+g|^p dx) ≤ (∫|f|^p dx) + (∫|g|^p dx)
D) (∫|f+g|^p dx)^(p) ≤ (∫|f|^p dx)^(p) + (∫|g|^p dx)^(p)
20. If x ∈ ℓp and y ∈ ℓp, what can we say about the sequence z_n = x_n * y_n?
A) z ∈ ℓp
B) z ∈ ℓp/2
C) z ∈ ℓ1
D) z ∈ ℓp'
21. What is the Hölder conjugate of p = ∞?
A) ∞
B) 1
C) 0
D) Undefined
22. What is the Hölder conjugate of p = 1?
A) 1
B) ∞
C) 0
D) Undefined
23. Is ℓ∞ separable?
A) Yes
B) No
C) Only if the underlying field is finite
D) Depends on the dimension
24. The space ℓp is separable for which values of p?
A) p = 1
B) p = 2
C) 1 ≤ p < ∞
D) All p
25. What is the ℓ∞ norm of the sequence (1, 2, 3, ..., N)?
A) N
B) N^2
C) N(N+1)/2
D) 1
26. If x ∈ ℓp and c is a scalar, what is ||cx||_p?
A) c ||x||_p
B) |c| ||x||_p
C) c^p ||x||_p
D) |c|^p ||x||_p
27. What is the ℓp norm of the sequence (1, 1, ..., 1) (finite sequence of length N)?
A) N
B) N^(1/p)
C) N^p
D) 1
28. For which p is the sequence (1/n)_{n=1}^∞ an element of ℓp?
A) p = 1
B) p = 2
C) p > 1
D) p > 2
29. Which of the following sequences is NOT in ℓ1?
A) (1, 1/2, 1/4, 1/8, ...)
B) (1, 1, 1/2, 1/3, ...)
C) (1/n^2)_{n=1}^∞
D) (1/n)_{n=1}^∞
30. What does the statement 'ℓp is a sequence space' mean?
A) It is a space of functions
B) It is a space of convergent sequences
C) It is a space of sequences of real or complex numbers that satisfy a certain norm condition
D) It is a space of matrices
31. The Cauchy-Schwarz inequality is a special case of the Hölder inequality when:
A) p = 1
B) p = 2
C) p = ∞
D) p = 1/2
32. If x ∈ ℓp and y ∈ ℓq with 1 ≤ p, q ≤ ∞, under what condition is the sequence x * y (term-wise product) in ℓr?
A) 1/p + 1/q = 1/r
B) 1/p + 1/q < 1/r
C) 1/p + 1/q > 1/r
D) p + q = r
33. Consider x = (1, 1, 1, ...) and y = (1, 1, 1, ...). For which p does ∑_{n=1}^∞ |x_n y_n| diverge?
A) p = 1
B) p = 2
C) p = ∞
D) Never
34. What is the result of the sum ∑_{n=1}^∞ |x_n y_n| when x = (1, 0, 0, ...) and y = (0, 1, 0, ...)?
A) 0
B) 1
C) ∞
D) Undefined
35. What is the condition for equality in the Hölder inequality ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p ||y||_p'?
A) x = 0 or y = 0
B) There is no condition for equality
C) The sequences ( |x_n|^p ) and ( |y_n|^p' ) are linearly dependent
D) The sequences ( |x_n| ) and ( |y_n| ) are proportional
36. For ℓ∞, does the Minkowski inequality hold in the form ||x + y||_∞ ≤ ||x||_∞ + ||y||_∞?
A) No, it only holds for finite sums
B) Yes, it holds
C) No, it holds in the form ||x + y||_∞ ≥ ||x||_∞ + ||y||_∞
D) No, it holds in the form ||x + y||_∞ = max(||x||_∞, ||y||_∞)
37. The Minkowski inequality is a generalization of which fundamental property of norms?
A) Homogeneity
B) Positive definiteness
C) Triangle inequality
D) Subadditivity
38. What is the Minkowski inequality for elements x and y in ℓp, 1 ≤ p < ∞?
A) ||x + y||_p ≤ ||x||_p + ||y||_p
B) ||x + y||_p ≥ ||x||_p + ||y||_p
C) ||x + y||_p = ||x||_p + ||y||_p
D) ||x + y||_p ≤ ||x||_p * ||y||_p
39. Which inequality is used to prove that ℓp spaces are complete (i.e., are Banach spaces)?
A) Cauchy-Schwarz inequality
B) Hölder inequality
C) Minkowski inequality
D) Triangle inequality
40. What is the result of applying Hölder's inequality when p = 2?
A) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_2 ||y||_2
B) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_1 ||y||_∞
C) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_2 ||y||_1
D) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_∞ ||y||_2
41. What is the Hölder inequality for two sequences x = (x_n) and y = (y_n) in ℓp and ℓp' respectively?
A) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p ||y||_p'
B) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p + ||y||_p'
C) ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p / ||y||_p'
D) ∑_{n=1}^∞ |x_n y_n| ≤ max(||x||_p, ||y||_p')
42. What is the condition that must be satisfied by p and its Hölder conjugate p'?
A) p + p' = 1
B) p * p' = 1
C) 1/p + 1/p' = 1
D) p - p' = 1
43. What is the Hölder conjugate of p, denoted by p'?
A) p' = p
B) p' = 1/p
C) p' = p / (p - 1)
D) p' = 1 - p
44. The ℓp spaces are a type of:
A) Banach space
B) Metric space
C) Topological space
D) Normed linear space
45. For which value of p is ℓp a Hilbert space?
A) p = 1
B) p = 2
C) p = ∞
D) All values of p
46. Consider the sequence x = (1, 1, 0, 0, ...). What is ||x||_p for any 1 ≤ p < ∞?
A) ∞
B) 1
C) 2^(1/p)
D) 2
47. Which of the following is an example of an element in ℓ2?
A) (1, 1/2, 1/3, 1/4, ...)
B) (1, 2, 3, 4, ...)
C) (1, 1, 1, 1, ...)
D) (1, 0, 0, 0, ...)
48. What is the definition of the norm for an element x in the ℓ∞ space?
A) ||x||_∞ = (∑_{n=1}^∞ |x_n|^∞)^(1/∞)
B) ||x||_∞ = ∑_{n=1}^∞ |x_n|
C) ||x||_∞ = sup_{n} |x_n|
D) ||x||_∞ = (∑_{n=1}^∞ |x_n|^2)^(1/2)
49. What is the definition of the norm for an element x in the ℓp space, 1 ≤ p < ∞?
A) ||x||_p = (∑_{n=1}^∞ |x_n|^p)^(1/p)
B) ||x||_p = sup_{n} |x_n|
C) ||x||_p = ∑_{n=1}^∞ |x_n|^p
D) ||x||_p = (∑_{n=1}^∞ |x_n|)^(1/p)
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