TRB PG Assistant - ℓp spaces - Hölder and Minkowski inequalities One Line Questions & Answers

1. The Minkowski inequality can be written in integral form for L^p spaces as: — (∫|f+g|^p dx)^(1/p) ≤ (∫|f|^p dx)^(1/p) + (∫|g|^p dx)^(1/p)

2. Which of the following is an example of an element in ℓ2? — (1, 1/2, 1/3, 1/4, ...)

3. Which of the following sequences is NOT in ℓ1? — (1/n)_{n=1}^∞

4. If p = 1, what is the Hölder inequality ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_1 ||y||_∞? — ∑_{n=1}^∞ |x_n y_n| ≤ (∑_{n=1}^∞ |x_n|) * (sup_{n} |y_n|)

5. What is the result of applying Hölder's inequality when p = 2? — ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_2 ||y||_2

6. What is the Hölder inequality for two sequences x = (x_n) and y = (y_n) in ℓp and ℓp' respectively? — ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p ||y||_p'

7. What is the Minkowski inequality for elements x and y in ℓp, 1 ≤ p < ∞? — ||x + y||_p ≤ ||x||_p + ||y||_p

8. What is the definition of the norm for an element x in the ℓ∞ space? — ||x||_∞ = sup_{n} |x_n|

9. What is the definition of the norm for an element x in the ℓp space, 1 ≤ p < ∞? — ||x||_p = (∑_{n=1}^∞ |x_n|^p)^(1/p)

10. Which of the following is NOT a property of the ℓp norm for 1 ≤ p < ∞? — ||cx||_p = c ||x||_p for any scalar c

11. Consider the sequence x = (1, 1, 0, 0, ...). What is ||x||_p for any 1 ≤ p < ∞? — 1

12. What is the Hölder conjugate of p = ∞? — 1

13. What is the result of the sum ∑_{n=1}^∞ |x_n y_n| when x = (1, 0, 0, ...) and y = (0, 1, 0, ...)? — 0

14. What is the Hölder conjugate of p = 1? — ∞

15. What is the ℓ2 norm of the sequence (1, 1, 1, ...)? — ∞

16. If x ∈ ℓp and y ∈ ℓq with 1 ≤ p, q ≤ ∞, under what condition is the sequence x * y (term-wise product) in ℓr? — 1/p + 1/q = 1/r

17. What is the ℓ1 norm of the sequence (1, 1/2, 1/4, 1/8, ...)? — 2

18. What is the ℓp norm of the sequence x = (2, 0, 0, 0, ...)? — 2^(1/p)

19. The ℓp spaces are a type of: — Banach space

20. If x ∈ ℓp and c is a scalar, what is ||cx||_p? — |c| ||x||_p

21. Which inequality is used to prove that ℓp spaces are complete (i.e., are Banach spaces)? — Minkowski inequality

22. If x ∈ ℓp and y ∈ ℓq such that 1/p + 1/q = 1, then the dot product ∑ x_n y_n is: — Defined and bounded by ||x||_p ||y||_q

23. Which inequality is fundamental for proving that the sum of two elements in ℓp is also in ℓp? — Minkowski's inequality

24. The Minkowski inequality is a generalization of which fundamental property of norms? — Triangle inequality

25. What is the significance of Hölder's inequality in the context of ℓp spaces? — It relates the norms of sequences under term-wise multiplication

26. What is the significance of Minkowski's inequality in the context of ℓp spaces? — It establishes the triangle inequality for the norm

27. What does the statement 'ℓp is a sequence space' mean? — It is a space of sequences of real or complex numbers that satisfy a certain norm condition

28. The dual space of ℓ1 is isomorphic to: — ℓ∞

29. The dual space of ℓ∞ is isomorphic to: — ℓ1

30. The dual space of ℓp for 1 < p < ∞ is isomorphic to: — ℓp'

31. What is the relationship between ℓp and ℓq when p < q? — No strict inclusion holds in general for infinite sequences

32. What is the ℓp norm of the sequence (1, 1, ..., 1) (finite sequence of length N)? — N^(1/p)

33. What is the ℓ∞ norm of the sequence (1, 2, 3, ..., N)? — N

34. For ℓ∞, does the Minkowski inequality hold in the form ||x + y||_∞ ≤ ||x||_∞ + ||y||_∞? — Yes, it holds

35. Consider the sequence x = (a, a, a, ...). For which p is x in ℓp? — Never if a ≠ 0 and the sequence is infinite

36. What is the condition that must be satisfied by p and its Hölder conjugate p'? — 1/p + 1/p' = 1

37. For which value of p is ℓp a Hilbert space? — p = 2

38. Consider x = (1, 1, 1, ...) and y = (1, 1, 1, ...). For which p does ∑_{n=1}^∞ |x_n y_n| diverge? — p = 1

39. The Cauchy-Schwarz inequality is a special case of the Hölder inequality when: — p = 2

40. For which p is the sequence (1/n)_{n=1}^∞ an element of ℓp? — p > 2

41. The space ℓp is separable for which values of p? — 1 ≤ p < ∞

42. What is the value of p for which ℓp and ℓp' are the same space (up to isomorphism)? — p = 2

43. What is the Hölder conjugate of p, denoted by p'? — p' = p / (p - 1)

44. Consider the sequence x = (1, 1, 1, ...). Which statement is true about its membership in ℓp? — x ∈ ℓp only for p = ∞

45. If x ∈ ℓp and ||x||_p = 0, what can we conclude about the sequence x? — x is the zero sequence (0, 0, 0, ...)

46. What is the condition for equality in the Hölder inequality ∑_{n=1}^∞ |x_n y_n| ≤ ||x||_p ||y||_p'? — The sequences ( |x_n|^p ) and ( |y_n|^p' ) are linearly dependent

47. Is ℓ∞ separable? — No

48. If x ∈ ℓp and y ∈ ℓp, what can we say about the sequence z_n = x_n * y_n? — z ∈ ℓ1

49. Consider x = (1, 1/2, 1/3, ...) and y = (1, 1/2, 1/3, ...). What is ∑_{n=1}^∞ x_n y_n? — ζ(2)

ℓp spaces - Hölder and Minkowski inequalities - One Line Questions