Banach spaces - definitions and examples, continuous linear transformations, Banach theorem, natural embedding of X in X'' - Question Bank

1. Which of the following statements about the dual space X* is FALSE?
A) If X is a normed space, X* is a normed space.
B) If X is a Banach space, X* is a Banach space.
C) If X is a reflexive Banach space, X* is reflexive.
D) If X is separable, X* is separable.
2. The Banach space X is called separable if it contains a:
A) Finite subset whose span is X.
B) Countable dense subset.
C) Complete orthonormal basis.
D) Closed subspace that is reflexive.
3. Which property is NOT guaranteed for the natural embedding J: X -> X'' for any normed space X?
A) Linearity
B) Continuity
C) Norm preservation (isometry)
D) Mapping X into X''
4. The Banach theorem (Uniform Boundedness Principle) is a powerful tool in functional analysis because it allows one to deduce uniform boundedness of a family of operators from:
A) Pointwise boundedness and completeness of the domain space.
B) Pointwise convergence and completeness of the codomain space.
C) Uniform convergence and completeness of the domain space.
D) Pointwise boundedness and completeness of the codomain space.
5. The natural embedding J: X -> X'' maps x to the functional J(x) defined by J(x)(f) = f(x) for f in X*. This definition ensures that J is:
A) An isometry
B) A linear map
C) A surjective map
D) A compact map
6. If T: X -> Y is a continuous linear operator between Banach spaces X and Y, and T is bounded, what can be said about the operator T:**: Y* -> X*?
A) T** is also continuous and bounded.
B) T** is continuous but not necessarily bounded.
C) T** is bounded but not necessarily continuous.
D) T** is neither continuous nor bounded.
7. Which condition is NOT equivalent to X being a reflexive Banach space?
A) The natural embedding J: X -> X'' is surjective.
B) The natural embedding J: X -> X'' is an isometry.
C) X is linearly isometric to its bidual X''.
D) X is homeomorphic to its bidual X''.
8. The concept of reflexivity in Banach spaces is related to the identification of the space with its:
A) Dual space
B) Second dual space via the natural embedding
C) Completion
D) Quotient space
9. Consider a sequence of linear operators T_n: X -> Y, where X is a Banach space. If T_n are continuous and sup_n ||T_n(x)|| < infinity for all x, what can be said about ||T_n||?
A) sup_n ||T_n|| < infinity
B) inf_n ||T_n|| < infinity
C) lim_n ||T_n|| = infinity
D) lim_n ||T_n|| = 0
10. Let T: X -> Y be a continuous linear operator between Banach spaces X and Y. The graph of T, G(T) = {(x, T(x)) : x in X}, is a closed subspace of X x Y. This is a consequence of:
A) The Banach theorem
B) The Open Mapping Theorem
C) The Closed Graph Theorem
D) The Hahn-Banach Theorem
11. Which theorem is crucial for proving that the dual of a Banach space is a Banach space?
A) Hahn-Banach Theorem
B) Uniform Boundedness Principle
C) Open Mapping Theorem
D) Cauchy-Schwarz Inequality
12. If X is a normed space, which of the following is always true about the relationship between X, X*, and X**?
A) X is always complete.
B) X* is always complete.
C) X** is always complete.
D) X, X*, and X** are all always complete.
13. The natural embedding map J: X -> X'' preserves norms, meaning ||J(x)|| = ||x|| for all x in X. This makes J:
A) A surjective map
B) An isometry
C) A compact map
D) A projection
14. Let X be a Banach space and Y be a normed space. If {T_n} is a sequence of continuous linear operators from X to Y such that sup_n ||T_n|| = infinity, does this imply that T_n(x) is unbounded for some x?
A) Yes, directly from the definition of the operator norm.
B) No, the Banach theorem is about pointwise boundedness.
C) Yes, by the Banach theorem (Uniform Boundedness Principle).
D) No, it implies the sequence T_n converges to 0.
15. The space l_infinity is a Banach space. Is it reflexive?
A) Yes
B) No
C) Only if it's finite dimensional
D) It depends on the field (real or complex)
16. The space l_p for p=1 is a Banach space. Is it reflexive?
A) Yes
B) No
C) Only if it's finite dimensional
D) It depends on the field (real or complex)
17. What is the definition of a Cauchy sequence in a normed vector space?
A) For every epsilon > 0, there exists an N such that ||x_n - x_m|| < epsilon for all n, m > N.
B) For every epsilon > 0, there exists an N such that ||x_n - x|| < epsilon for all n > N, for some x.
C) For every epsilon > 0, there exists an N such that ||x_n|| < epsilon for all n > N.
D) For every epsilon > 0, there exists an N such that ||x_n - x_{n+1}|| < epsilon for all n > N.
18. Let X be a Banach space and Y be a normed space. If T: X -> Y is a linear operator such that T(x_n) -> 0 whenever x_n -> 0 (i.e., T is continuous), then T is bounded.
A) True, by the Closed Graph Theorem.
B) False, continuity does not imply boundedness.
C) True, by the Open Mapping Theorem.
D) False, this statement is incorrect.
19. Which of the following is a key application of the Uniform Boundedness Principle (Banach Theorem)?
A) Proving the existence of solutions to differential equations.
B) Showing that a sequence of functions converging pointwise on a complete metric space is uniformly convergent.
C) Establishing the completeness of the dual space.
D) Characterizing compact operators.
20. If X is a normed space, when is X reflexive?
A) When X is separable
B) When the natural embedding J: X -> X'' is surjective
C) When X* is separable
D) When X is finite-dimensional
21. The natural embedding J: X -> X'' is always a linear transformation.
A) True
B) False
C) True only if X is finite dimensional
D) False
22. Consider the space C([0,1]) with the supremum norm. This space is a Banach space. What is its dual space C([0,1])*?
A) l_1([0,1])
B) The space of finite signed measures on [0,1]
C) C([0,1])
D) l_infinity([0,1])
23. Let X be a Banach space. The dual space X* is also a Banach space under the operator norm. This is a fundamental result.
A) True
B) False
C) True only if X is reflexive
D) False, X* is only a normed space
24. The Closed Graph Theorem states that if T: X -> Y is a linear operator between Banach spaces X and Y, and its graph is closed, then T is:
A) Continuous
B) Bounded
C) Compact
D) Surjective
25. If T: X -> Y is a bounded linear operator between Banach spaces X and Y, and T is surjective, then T is an open map. This is a statement of:
A) The Banach theorem
B) The Hahn-Banach theorem
C) The Open Mapping Theorem
D) The Closed Graph Theorem
26. The Open Mapping Theorem, closely related to the Banach theorem, states that a continuous linear surjective map between two Banach spaces is:
A) An isometry
B) A compact operator
C) An isomorphism
D) A projection
27. Which of these spaces is NOT reflexive?
A) Finite-dimensional normed spaces
B) l_p for 1 < p < infinity
C) l_1
D) l_2
28. What condition is required for the natural embedding J: X -> X'' to be an isometry?
A) X must be finite-dimensional.
B) X must be a Banach space.
C) X must be reflexive.
D) No additional condition is needed; it is always an isometry.
29. For a normed space X, the second dual X'' is always:
A) A normed space
B) A Banach space
C) A Hilbert space
D) A finite-dimensional space
30. The natural embedding J: X -> X'' maps an element x in X to:
A) A linear functional on X'
B) A linear functional on X
C) A linear operator on X'
D) An element in X'
31. If X is a Banach space and Y is a normed space, and {T_n} is a sequence of continuous linear operators from X to Y such that T_n(x) converges for each x in X, then T_n converges uniformly.
A) False, pointwise convergence does not imply uniform convergence.
B) True, by the Uniform Boundedness Principle.
C) True, if Y is also a Banach space.
D) False, this is a property of compact operators.
32. The Banach theorem is also known as:
A) Hahn-Banach Theorem
B) Riesz Representation Theorem
C) Uniform Boundedness Principle
D) Open Mapping Theorem
33. What is the definition of the norm of a continuous linear functional f in X*?
A) ||f|| = sup { |f(x)| : ||x|| = 1 }
B) ||f|| = inf { |f(x)| : ||x|| = 1 }
C) ||f|| = sup { |f(x)| : ||x|| < 1 }
D) ||f|| = inf { |f(x)| : ||x|| < 1 }
34. The set of all continuous linear functionals on a normed vector space X forms a normed vector space itself, called the:
A) Pre-dual space
B) Second dual space
C) Dual space
D) Completion of X
35. What is a continuous linear functional on a normed vector space X?
A) A linear map f: X -> R (or C) such that |f(x)| <= M||x|| for some M.
B) A linear map f: X -> R (or C) such that f(x+y) = f(x) + f(y).
C) A linear map f: X -> R (or C) such that f(ax) = af(x).
D) A linear map f: X -> R (or C) such that f(x) = 0 if x = 0.
36. Which of the following is a standard example of a reflexive Banach space?
A) C[0, 1]
B) l_1
C) l_2
D) l_infinity
37. If X is a reflexive Banach space, what property does the natural embedding J: X -> X'' have?
A) It is surjective.
B) It is an isometry.
C) It is a homeomorphism.
D) It is a compact operator.
38. The natural embedding J: X -> X'' is always:
A) Surjective
B) Injective
C) Bijective
D) An isometry
39. What is the norm of the natural embedding map J: X -> X''?
A) ||J|| = 1
B) ||J|| = 0
C) ||J|| = infinity
D) ||J|| depends on the dimension of X
40. What is the natural embedding map J: X -> X''?
A) J(x) = f_x, where f_x(g) = g(x) for all g in X'.
B) J(x) = g_x, where g_x(f) = f(x) for all f in X'.
C) J(x) = h_x, where h_x(T) = T(x) for all T in L(X).
D) J(x) = k_x, where k_x(y) = <x, y> for all y in X.
41. Let X be a normed vector space. What is the space X''?
A) The dual space of X.
B) The second dual space of X, consisting of continuous linear functionals on X'.
C) The completion of X.
D) The space of all bounded linear operators on X.
42. Consider a sequence of continuous linear operators {T_n} from a Banach space X to a normed space Y such that sup_n ||T_n(x)|| < infinity for every x in X. The Banach theorem states that:
A) sup_n ||T_n|| < infinity
B) inf_n ||T_n|| < infinity
C) lim_n ||T_n|| = 0
D) lim_n ||T_n|| = infinity
43. What does the Banach theorem imply about a pointwise bounded sequence of continuous linear operators from a Banach space X to a normed space Y?
A) The sequence is uniformly bounded.
B) The sequence is uniformly continuous.
C) The sequence is uniformly convergent.
D) The sequence is uniformly compact.
44. The Banach theorem, also known as the Uniform Boundedness Principle, applies to a collection of operators between which types of spaces?
A) Two arbitrary metric spaces
B) A complete metric space and a normed vector space
C) A complete normed vector space and a normed vector space
D) Two arbitrary normed vector spaces
45. If T: X -> Y is a bounded linear transformation, what is the norm of T, denoted ||T||?
A) The smallest M such that ||T(x)|| <= M||x|| for all x.
B) The largest M such that ||T(x)|| <= M||x|| for all x.
C) The supremum of ||T(x)|| over all x with ||x|| = 1.
D) The infimum of ||T(x)|| over all x with ||x|| = 1.
46. What is a linear transformation T: X -> Y between normed spaces X and Y called if there exists a constant M such that ||T(x)|| <= M||x|| for all x in X?
A) Bounded linear transformation
B) Unbounded linear transformation
C) Isomorphic linear transformation
D) Compact linear transformation
47. In a normed vector space X, what does completeness mean?
A) Every Cauchy sequence converges to an element within X.
B) Every convergent sequence converges to an element within X.
C) Every sequence has a convergent subsequence.
D) Every bounded sequence is convergent.
48. Which of the following is NOT a standard example of a Banach space?
A) The space of continuous real-valued functions on a compact interval C[a, b]
B) The space of all bounded sequences of real numbers l_infinity
C) The space of all square-summable sequences l_2
D) The space of all polynomials on [0, 1] P[0, 1]
49. What is the fundamental property of a Banach space?
A) It is a finite-dimensional normed vector space.
B) It is a complete normed vector space.
C) It is a vector space with an inner product.
D) It is a topological vector space with a countable basis.
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