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Hilbert Spaces

This quiz will test your understanding of the concepts related to Hilbert Spaces, a fundamental topic in functional analysis.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

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.
Question 2 Multiple Choice (Single Answer)

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. $\{\frac{1}{\sqrt{n}} \sin(n\pi x) \mid n \in \mathbb{N}\}$
Question 3 Multiple Choice (Single Answer)

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.
Question 4 Multiple Choice (Single Answer)

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.
Question 5 Multiple Choice (Single Answer)

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.
Question 6 Multiple Choice (Single Answer)

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.
Question 7 Multiple Choice (Single Answer)

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.
Question 8 Multiple Choice (Single Answer)

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.
Question 9 Multiple Choice (Single Answer)

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.
Question 10 Multiple Choice (Single Answer)

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.
Question 11 Multiple Choice (Single Answer)

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.
Question 12 Multiple Choice (Single Answer)

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.
Question 13 Multiple Choice (Single Answer)

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.
Question 14 Multiple Choice (Single Answer)

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.
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