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Functional Analysis

This quiz covers the fundamental concepts and theorems of Functional Analysis, a branch of mathematics that deals with the study of function spaces and linear operators acting on them.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a complete normed space?

  1. (C[0, 1], ||.||_2)
  2. (C[0, 1], ||.||_∞)
  3. (L^2[0, 1], ||.||_2)
  4. (L^1[0, 1], ||.||_1)
Question 2 Multiple Choice (Single Answer)

Which of the following is a Banach space?

  1. (C[0, 1], ||.||_∞)
  2. (L^1[0, 1], ||.||_1)
  3. (L^2[0, 1], ||.||_2)
  4. (C[0, 1], ||.||_2)
Question 3 Multiple Choice (Single Answer)

Which of the following is a Hilbert space?

  1. (C[0, 1], ||.||_2)
  2. (L^2[0, 1], ||.||_2)
  3. (L^1[0, 1], ||.||_1)
  4. (C[0, 1], ||.||_∞)
Question 4 Multiple Choice (Single Answer)

Which of the following is an example of a bounded linear operator?

  1. The derivative operator on C^1[0, 1]
  2. The integration operator on L^1[0, 1]
  3. The multiplication operator on L^2[0, 1]
  4. The shift operator on C[0, 1]
Question 5 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 integral operator on L^2[0, 1]
  3. The derivative operator on C^1[0, 1]
  4. The shift operator on C[0, 1]
Question 6 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 dual space.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Banach space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 7 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 element of the space.
  2. Every closed subspace of a Hilbert space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 8 Multiple Choice (Single Answer)

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

  1. Every compact self-adjoint operator on a Hilbert space has a pure point spectrum.
  2. Every bounded linear operator on a Hilbert space has a bounded inverse.
  3. Every closed subspace of a Hilbert space has a complement.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 9 Multiple Choice (Single Answer)

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

  1. Every bounded linear operator on a Banach space is open.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 10 Multiple Choice (Single Answer)

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

  1. Every bounded linear operator on a Banach space is closed.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 11 Multiple Choice (Single Answer)

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

  1. Every bounded sequence of linear operators on a Banach space is uniformly bounded.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 12 Multiple Choice (Single Answer)

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

  1. Every closed convex subset of a locally compact Hausdorff space is the closure of its extreme points.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 13 Multiple Choice (Single Answer)

Which of the following is a consequence of the Mazur's theorem?

  1. Every separable Banach space is isometrically isomorphic to a closed subspace of C[0, 1].
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 14 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 subset of a Banach space has a fixed point.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
Question 15 Multiple Choice (Single Answer)

Which of the following is a consequence of the Tychonoff's theorem?

  1. Every product of compact Hausdorff spaces is compact.
  2. Every closed subspace of a Banach space has a complement.
  3. Every bounded linear operator on a Hilbert space has a bounded inverse.
  4. Every compact operator on a Hilbert space has a pure point spectrum.
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