TRB PG Assistant - Hilbert spaces - orthonormal bases, conjugate space, adjoint of an operator, projections Online Test (MCQ Quiz)

1. What is the defining characteristic of a Hilbert space?

A) It is a finite-dimensional vector space with a norm. B) It is a complete inner product space. C) It is a vector space with a metric. D) It is a vector space with a topology.

2. In a Hilbert space, what property must an orthonormal set satisfy?

A) The inner product of any two distinct vectors is 1. B) The inner product of any two distinct vectors is 0, and the norm of each vector is 1. C) The inner product of any two distinct vectors is 1, and the norm of each vector is 0. D) The inner product of any two distinct vectors is 0, and the norm of each vector is 0.

3. What is the significance of an orthonormal basis in a Hilbert space?

A) It allows for any vector to be represented as a finite sum of basis vectors. B) It allows for any vector to be represented as an infinite series of projections onto the basis vectors. C) It simplifies the computation of distances but not inner products. D) It is only possible in finite-dimensional Hilbert spaces.

4. Bessel's inequality for an orthonormal set {e_i} in a Hilbert space H and any vector x in H states:

A) Sum(|<x, e_i>|^2) >= ||x||^2 B) Sum(|<x, e_i>|^2) <= ||x||^2 C) Sum(|<x, e_i>|) <= ||x|| D) Sum(|<x, e_i>|) >= ||x||

5. When does Bessel's inequality become Parseval's identity for a complete orthonormal set {e_i} in a Hilbert space H and a vector x in H?

A) When the sum is finite. B) When the set is countable and the sum converges to ||x||^2. C) When the set is uncountable. D) When x is orthogonal to all e_i.

6. What is the conjugate space (dual space) of a Hilbert space H, denoted by H*?

A) The space of all linear operators on H. B) The space of all bounded linear functionals on H. C) The space of all continuous functions on H. D) The space of all self-adjoint operators on H.

7. The Riesz Representation Theorem establishes a canonical isomorphism between a Hilbert space H and its conjugate space H*. What does this theorem state?

A) Every bounded linear functional f in H* corresponds to a unique vector y in H such that f(x) = <x, y> for all x in H. B) Every vector y in H corresponds to a unique bounded linear functional f in H* such that f(x) = <x, y> for all x in H. C) Every bounded linear operator T on H corresponds to a unique vector y in H such that T(x) = <x, y>. D) Every vector y in H corresponds to a unique bounded linear operator T on H such that T(x) = <x, y>.

8. For a bounded linear operator T from a Hilbert space H1 to a Hilbert space H2, what is the conjugate space of H1?

A) H1 itself. B) H2. C) The space of bounded linear functionals on H1 (H1*). D) The space of bounded linear functionals on H2 (H2*).

9. Let T be a bounded linear operator from Hilbert space H1 to Hilbert space H2. The adjoint operator, denoted by T*, maps from which space to which space?

A) H1 to H1 B) H2 to H1 C) H1* to H2* D) H2 to H1

10. The adjoint operator T* of a bounded linear operator T: H1 -> H2 is defined by the relation:

A) <T(x), y> = <x, T*(y)> for all x in H1, y in H2 B) <T(x), y> = <T*(x), y> for all x in H1, y in H2 C) <x, T(y)> = <T*(x), y> for all x in H1, y in H2 D) <T(x), y> = <x, T(y)> for all x in H1, y in H2

11. If T is a bounded linear operator on a Hilbert space H, what is the relationship between T** and T?

A) T** = -T B) T** = T C) T** = T* D) T** is the inverse of T

12. An operator T on a Hilbert space H is called self-adjoint if:

A) T* = -T B) T* = T C) T* = T^-1 D) T* = 0

13. What property does a projection operator P on a Hilbert space satisfy?

A) P* = P and P^2 = P B) P* = -P and P^2 = P C) P* = P and P^2 = 0 D) P* = P and P = I (identity operator)

14. A projection operator P in a Hilbert space projects onto a subspace. What kind of subspace is it?

A) Any arbitrary subspace. B) A closed subspace. C) A finite-dimensional subspace. D) A dense subspace.

15. If P is a projection onto a closed subspace M of a Hilbert space H, then for any x in H, P(x) is the vector in M that minimizes the distance to x. This means:

A) ||x - P(x)|| <= ||x - m|| for all m in M B) ||x - P(x)|| >= ||x - m|| for all m in M C) ||P(x)|| = ||x|| D) P(x) = x

16. Let P1 and P2 be two projection operators on a Hilbert space H. When is their sum P1 + P2 also a projection?

A) Always. B) If P1 * P2 = 0 (i.e., their ranges are orthogonal). C) If P1 * P2 = P1. D) If P1 and P2 commute (P1 * P2 = P2 * P1).

17. Let P1 and P2 be two projection operators on a Hilbert space H. When is their product P1 * P2 also a projection?

A) Always. B) If P1 and P2 commute (P1 * P2 = P2 * P1). C) If P1 * P2 = 0. D) If P1 = P2.

18. The set of all bounded linear operators on a Hilbert space H forms an algebra, which is a Banach algebra. What is the norm in this algebra?

A) The operator norm. B) The trace norm. C) The Hilbert-Schmidt norm. D) The spectral norm.

19. What is the conjugate space of the space L^2([a, b])?

A) L^2([a, b]) itself. B) L^1([a, b]). C) C([a, b]). D) The space of all measurable functions on [a, b].

20. Consider the Hilbert space l^2 of square-summable sequences. Let T be an operator such that T((x_n)) = (0, x_1, x_2, ...). What is T*?

A) T*((y_n)) = (y_2, y_3, ...) B) T*((y_n)) = (x_2, x_3, ...) C) T*((y_n)) = (y_1, y_0, y_1, ...) D) T*((y_n)) = (0, y_1, y_2, ...)

21. If T is a bounded linear operator on a Hilbert space H, then the adjoint T* satisfies ||T*|| = ||T||. This implies that the algebra of bounded linear operators is:

A) A Hilbert algebra. B) A C*-algebra. C) A commutative algebra. D) An associative algebra.

22. What is the spectral theorem for self-adjoint operators on a Hilbert space about?

A) Representing self-adjoint operators as limits of projections. B) Representing self-adjoint operators as multiplication operators. C) Representing self-adjoint operators as diagonal matrices in some basis. D) All of the above.

23. A Hilbert space is separable if it contains a:

A) Finite orthonormal set. B) Countable dense subset. C) Uncountable dense subset. D) Compact subset.

24. The Gram-Schmidt process is used to construct:

A) A Hilbert space from a normed space. B) An orthonormal basis from any linearly independent set. C) The adjoint of an operator. D) A projection operator.

25. Which of the following is NOT a Hilbert space?

A) R^n with the standard dot product. B) L^2([0, 1]) with the integral of the product. C) C([0, 1]) with the sup norm. D) l^2, the space of square-summable sequences.

26. If T is a bounded linear operator on a Hilbert space H, and T is normal (i.e., T*T = TT*), then:

A) T is self-adjoint. B) T is unitary. C) T is diagonalizable by a unitary operator. D) T is a projection.

27. What is the kernel (null space) of a projection operator P onto a closed subspace M?

A) M itself. B) The orthogonal complement of M (M^perp). C) The entire Hilbert space H. D) The zero vector {0}.

28. Let {e_n} be an orthonormal basis for a separable Hilbert space H. For any vector x in H, the Fourier coefficients are given by:

A) c_n = ||x|| * ||e_n|| B) c_n = <x, e_n> C) c_n = ||x|| / ||e_n|| D) c_n = <x, x> * <e_n, e_n>

29. The conjugate space of a finite-dimensional Hilbert space is:

A) Isomorphic to the space itself. B) Always the zero space. C) Isomorphic to the space of linear transformations. D) Isomorphic to the space of continuous functions.

30. If T is a bounded linear operator on a Hilbert space H, what is the definition of the operator norm ||T||?

A) sup { ||Tx|| : ||x|| = 1 } B) sup { |<Tx, x>| : ||x|| = 1 } C) sup { ||Tx|| : ||x|| <= 1 } D) sup { ||x|| : ||Tx|| = 1 }

31. What is the adjoint of the identity operator I on a Hilbert space H?

A) 0 (the zero operator) B) -I C) I D) I*

32. If T is a linear operator on a Hilbert space H such that <Tx, y> = <x, Ty> for all x, y in H, then T must be:

A) Anti-self-adjoint. B) Self-adjoint. C) Unitary. D) A projection.

33. A projection P onto a closed subspace M is orthogonal if and only if:

A) P = P* B) P^2 = P C) The range of P is M and the kernel of P is M^perp. D) The range of P is M and the kernel of P is {0}.

34. Consider the Hilbert space L^2([0, 1]). The operator of multiplication by x, (Mf)(t) = t*f(t), is:

A) Not a bounded operator. B) A bounded self-adjoint operator. C) A bounded normal operator but not self-adjoint. D) An unbounded operator.

35. If T is a bounded linear operator on a Hilbert space H, then the operator T*T is:

A) Always self-adjoint and positive semi-definite. B) Always unitary. C) Always a projection. D) Always normal but not necessarily self-adjoint.

36. What is the conjugate space of a finite-dimensional vector space V?

A) V itself. B) The dual space V*. C) The space of linear transformations on V. D) The space of bilinear forms on V.

37. For a Hilbert space H, the set of all bounded linear operators B(H) equipped with the operator norm and the adjoint operation forms:

A) A vector space. B) A C*-algebra. C) A Hilbert algebra. D) All of the above.

38. If T is a bounded linear operator on a Hilbert space H, and T is unitary, then:

A) T*T = I and TT* = I B) T* = T C) T* = -T D) T*T = 0

39. The concept of an orthonormal basis is crucial for:

A) Defining the Hilbert space structure. B) Representing vectors via Fourier series/coefficients. C) Simplifying operator theory. D) All of the above.

40. What is the adjoint of an operator T defined by <Tx, y> = x_1*y_2 + x_2*y_1 on R^2?

A) T* defined by <T*x, y> = x_1*y_2 + x_2*y_1 B) T* defined by <T*x, y> = x_2*y_1 + x_1*y_2 C) T* defined by <T*x, y> = x_1*y_1 + x_2*y_2 D) T* is the same as T.

41. In a Hilbert space, if {e_i} is an orthonormal set, then for any vector x, the projection of x onto the subspace spanned by {e_i} is given by:

A) Sum(<x, e_i>) * e_i B) Sum(<x, e_i>^2) * e_i C) Sum(<x, e_i>) * <x, e_i> D) Sum(<x, e_i>) * e_i^2

42. A projection P is called an orthogonal projection if its range and kernel are:

A) Equal. B) Disjoint. C) Orthogonal complements of each other. D) Identical.

43. The conjugate space H* of a Hilbert space H is itself a:

A) Normed vector space. B) Banach space. C) Hilbert space. D) All of the above.

44. If T is a bounded linear operator on a Hilbert space H, then the operator TT* is:

A) Always self-adjoint and positive semi-definite. B) Always unitary. C) Always a projection. D) Always normal but not necessarily self-adjoint.

45. What is the adjoint of the zero operator 0 on a Hilbert space H?

A) 0 B) I C) -0 D) 0*

46. Which theorem guarantees that every closed subspace of a Hilbert space is the range of some orthogonal projection?

A) Riesz Representation Theorem B) Spectral Theorem C) Projection Theorem D) Bessel's Inequality

47. If H is a Hilbert space and T is a bounded linear operator on H, then T is invertible if and only if:

A) T* is invertible. B) T*T is invertible. C) T is injective and surjective. D) All of the above.

48. The conjugate space of the space of continuous functions C([a, b]) with the sup norm is NOT necessarily a Hilbert space. This highlights the importance of the:

A) Inner product structure. B) Completeness property. C) Separability property. D) Boundedness property.

49. Consider an orthonormal basis {e_n} in a separable Hilbert space H. For any two vectors x and y, Parseval's identity for the inner product is:

A) <x, y> = Sum(<x, e_n>) * <y, e_n> B) <x, y> = Sum(<x, e_n>^2) * <y, e_n>^2 C) <x, y> = Sum(<x, e_n>) + Sum(<y, e_n>) D) <x, y> = Sum(|<x, e_n>| * |<y, e_n>|)

Hilbert spaces - orthonormal bases, conjugate space, adjoint of an operator, projections - Online Test

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