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Hilbert Spaces

Description: This quiz will test your understanding of the concepts related to Hilbert Spaces, a fundamental topic in functional analysis.
Number of Questions: 14
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Tags: hilbert spaces functional analysis inner product spaces linear algebra
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Let $H$ be a Hilbert space. Which of the following is NOT a property of the inner product $\langle \cdot, \cdot \rangle$ on $H$?

  1. It is linear in the first argument.

  2. It is conjugate linear in the second argument.

  3. It is positive definite.

  4. It is bounded.

Show answer
Correct Option: D
Explanation:

The inner product $\langle \cdot, \cdot \rangle$ on a Hilbert space is not necessarily bounded. However, it is always linear in the first argument, conjugate linear in the second argument, and positive definite.

Which of the following is a complete orthonormal set in the Hilbert space $L^2([0, 1])$?

  1. ${1, x, x^2, \ldots}$

  2. ${\sin(n\pi x), \cos(n\pi x) \mid n \in \mathbb{N}}$

  3. ${e^{inx} \mid n \in \mathbb{Z}}$

  4. ${\frac{1}{\sqrt{n}} \sin(n\pi x) \mid n \in \mathbb{N}}$

Show answer
Correct Option: B
Explanation:

The set ${\sin(n\pi x), \cos(n\pi x) \mid n \in \mathbb{N}}$ is a complete orthonormal set in $L^2([0, 1])$ because it is an orthonormal basis for the subspace of $L^2([0, 1])$ consisting of functions that are continuous on $[0, 1]$ and have a continuous derivative on $(0, 1)$.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Let $H$ be a Hilbert space and $T : H \rightarrow H$ be a bounded linear operator. Which of the following is NOT a property of the adjoint operator $T^*$?

  1. It is bounded.

  2. It is linear.

  3. It is conjugate linear.

  4. It is invertible if $T$ is invertible.

Show answer
Correct Option: D
Explanation:

The adjoint operator $T^$ is not necessarily invertible even if $T$ is invertible. For example, consider the shift operator $S : L^2([0, 1]) \rightarrow L^2([0, 1])$ defined by $Sf(x) = f(x-1)$. This operator is invertible, but its adjoint $S^$ is not invertible.

Let $H$ be a Hilbert space and $x, y \in H$. Which of the following is NOT a property of the inner product $\langle x, y \rangle$?

  1. It is a complex number.

  2. It is conjugate symmetric.

  3. It is positive definite.

  4. It is bounded.

Show answer
Correct Option: D
Explanation:

The inner product $\langle x, y \rangle$ is not necessarily bounded. However, it is always a complex number, conjugate symmetric, and positive definite.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

Which of the following is a Hilbert space?

  1. The space of continuous functions on $[0, 1]$ with the supremum norm.

  2. The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm.

  3. The space of polynomials with the $L^2$-norm.

  4. The space of bounded linear operators on a Hilbert space with the operator norm.

Show answer
Correct Option: B
Explanation:

The space of square-integrable functions on $[0, 1]$ with the $L^2$-norm is a Hilbert space because it is a complete inner product space.

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