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Functional Analysis and Operator Theory

This quiz covers fundamental concepts, theorems, and applications in Functional Analysis and Operator Theory.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a complete normed space?

  1. L^2([0, 1])
  2. C([0, 1])
  3. L^1([0, 1])
  4. C^1([0, 1])
Question 2 Multiple Choice (Single Answer)

Which of the following is an example of a compact operator?

  1. The identity operator on L^2([0, 1])
  2. The differentiation operator on C^1([0, 1])
  3. The integration operator on L^1([0, 1])
  4. The multiplication operator by x on L^2([0, 1])
Question 3 Multiple Choice (Single Answer)

Which of the following is a consequence of the Hahn-Banach theorem?

  1. Every linear functional on a normed space can be extended to a linear functional on its completion.
  2. Every bounded linear operator on a Banach space can be extended to a bounded linear operator on its dual space.
  3. Every closed subspace of a Hilbert space is complemented.
  4. Every self-adjoint operator on a Hilbert space has a spectral resolution.
Question 4 Multiple Choice (Single Answer)

Which of the following is a consequence of the Riesz representation theorem?

  1. Every bounded linear functional on a Hilbert space can be represented as an inner product with a unique vector in the space.
  2. Every closed subspace of a Hilbert space is complemented.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 5 Multiple Choice (Single Answer)

Which of the following is a consequence of the spectral theorem for self-adjoint operators?

  1. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  2. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
  3. Every bounded linear operator on a Banach space can be represented as a multiplication operator.
  4. Every closed subspace of a Hilbert space is complemented.
Question 6 Multiple Choice (Single Answer)

Which of the following is a consequence of the closed graph theorem?

  1. Every closed linear operator on a Banach space is bounded.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 7 Multiple Choice (Single Answer)

Which of the following is a consequence of the Banach-Alaoglu theorem?

  1. The unit ball of the dual space of a Banach space is weak*-compact.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 8 Multiple Choice (Single Answer)

Which of the following is a consequence of the Krein-Milman theorem?

  1. Every compact convex set in a locally convex space is the closed convex hull of its extreme points.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 9 Multiple Choice (Single Answer)

Which of the following is a consequence of the Schauder fixed-point theorem?

  1. Every continuous self-map of a compact convex set in a Banach space has a fixed point.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 10 Multiple Choice (Single Answer)

Which of the following is a consequence of the open mapping theorem?

  1. Every bounded linear operator with closed range from a Banach space to another Banach space is open.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 11 Multiple Choice (Single Answer)

Which of the following is a consequence of the uniform boundedness principle?

  1. If a sequence of bounded linear operators on a Banach space is pointwise bounded, then it is uniformly bounded.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 12 Multiple Choice (Single Answer)

Which of the following is a consequence of the Hahn-Banach separation theorem?

  1. Every two disjoint convex sets in a locally convex space can be separated by a hyperplane.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 13 Multiple Choice (Single Answer)

Which of the following is a consequence of the Stone-Weierstrass theorem?

  1. Every continuous function on a compact Hausdorff space can be uniformly approximated by a sequence of polynomials.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 14 Multiple Choice (Single Answer)

Which of the following is a consequence of the Riesz-Markov-Kakutani representation theorem?

  1. Every positive linear functional on a C*-algebra is represented by a unique positive measure on the spectrum of the C*-algebra.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
Question 15 Multiple Choice (Single Answer)

Which of the following is a consequence of the Gelfand-Naimark theorem?

  1. Every C*-algebra is isomorphic to a closed subalgebra of the algebra of bounded linear operators on a Hilbert space.
  2. Every bounded linear operator on a Banach space has a closed graph.
  3. Every self-adjoint operator on a Hilbert space has a spectral resolution.
  4. Every normal operator on a Hilbert space is unitarily equivalent to a multiplication operator.
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