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Functional Analysis

Description: 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.
Number of Questions: 15
Created by:
Tags: functional analysis normed spaces banach spaces hilbert spaces linear operators bounded linear operators compact operators
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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)

Show answer
Correct Option: C
Explanation:

The space (L^2[0, 1], ||.||_2) is a complete normed space, also known as the Hilbert space of square-integrable functions.

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)

Show answer
Correct Option: B
Explanation:

The space (L^1[0, 1], ||.||_1) is a Banach space, also known as the space of absolutely integrable functions.

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], ||.||_∞)

Show answer
Correct Option: B
Explanation:

The space (L^2[0, 1], ||.||_2) is a Hilbert space, which is a complete inner product space.

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]

Show answer
Correct Option: C
Explanation:

The multiplication operator on L^2[0, 1] is a bounded linear operator because it satisfies the inequality ||T(x)|| ≤ M ||x|| for some constant M and all x ∈ L^2[0, 1].

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]

Show answer
Correct Option: B
Explanation:

The integral operator on L^2[0, 1] is a compact operator because it maps bounded sets into relatively compact sets.

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.

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 dual space with 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 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.

Show answer
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 element of the space.

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.

Show answer
Correct Option: A
Explanation:

The spectral theorem for compact self-adjoint operators states that every compact self-adjoint operator on a Hilbert space has a pure point spectrum.

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.

Show answer
Correct Option: A
Explanation:

The open mapping theorem states that every bounded linear operator on a Banach space is open.

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.

Show answer
Correct Option: A
Explanation:

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

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.

Show answer
Correct Option: A
Explanation:

The Banach-Steinhaus theorem states that every bounded sequence of linear operators on a Banach space is uniformly bounded.

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.

Show answer
Correct Option: A
Explanation:

The Krein-Milman theorem states that every closed convex subset of a locally compact Hausdorff space is the closure of its extreme points.

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.

Show answer
Correct Option: A
Explanation:

Mazur's theorem states that every separable Banach space is isometrically isomorphic to a closed subspace of C[0, 1].

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.

Show answer
Correct Option: A
Explanation:

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

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.

Show answer
Correct Option: A
Explanation:

Tychonoff's theorem states that every product of compact Hausdorff spaces is compact.

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