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Functional Analysis and Operator Theory

Description: This quiz covers fundamental concepts, theorems, and applications in Functional Analysis and Operator Theory.
Number of Questions: 15
Created by:
Tags: functional analysis operator theory linear operators hilbert spaces banach spaces
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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])

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Correct Option: A
Explanation:

L^2([0, 1]) is a complete normed space because it is a Hilbert space.

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])

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Correct Option: D
Explanation:

The multiplication operator by x on L^2([0, 1]) is a compact operator because it is a bounded linear operator whose range is contained in a finite-dimensional subspace of L^2([0, 1]).

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.

Show answer
Correct Option: A
Explanation:

The Hahn-Banach theorem states that every linear functional on a normed space can be extended to a linear functional on its completion that has the same norm.

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.

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Correct Option: A
Explanation:

The Riesz representation theorem states that every bounded linear functional on a Hilbert space can be represented as an inner product with a unique vector in the space.

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.

Show answer
Correct Option: A
Explanation:

The spectral theorem for self-adjoint operators states that every self-adjoint operator on a Hilbert space has a spectral resolution.

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.

Show answer
Correct Option: A
Explanation:

The closed graph theorem states that every closed linear operator on a Banach space is bounded.

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.

Show answer
Correct Option: A
Explanation:

The Banach-Alaoglu theorem states that the unit ball of the dual space of a Banach space is weak*-compact.

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.

Show answer
Correct Option: A
Explanation:

The Krein-Milman theorem states that every compact convex set in a locally convex space is the closed convex hull of its extreme points.

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.

Show answer
Correct Option: A
Explanation:

The Schauder fixed-point theorem states that every continuous self-map of a compact convex set in a Banach space has a fixed point.

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.

Show answer
Correct Option: A
Explanation:

The open mapping theorem states that every bounded linear operator with closed range from a Banach space to another Banach space is open.

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.

Show answer
Correct Option: A
Explanation:

The uniform boundedness principle states that if a sequence of bounded linear operators on a Banach space is pointwise bounded, then it is uniformly bounded.

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.

Show answer
Correct Option: A
Explanation:

The Hahn-Banach separation theorem states that every two disjoint convex sets in a locally convex space can be separated by a hyperplane.

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.

Show answer
Correct Option: A
Explanation:

The Stone-Weierstrass theorem states that every continuous function on a compact Hausdorff space can be uniformly approximated by a sequence of polynomials.

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.

Show answer
Correct Option: A
Explanation:

The Riesz-Markov-Kakutani representation theorem states that every positive linear functional on a C*-algebra is represented by a unique positive measure on the spectrum of the C*-algebra.

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.

Show answer
Correct Option: A
Explanation:

The Gelfand-Naimark theorem states that every C*-algebra is isomorphic to a closed subalgebra of the algebra of bounded linear operators on a Hilbert space.

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