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Normed Vector Spaces

This quiz covers the fundamental concepts and properties of normed vector spaces, a cornerstone of functional analysis and linear algebra.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

In a normed vector space, the distance between two vectors x and y is defined as:

  1. ||x - y||
  2. ||x + y||
  3. ||x|| + ||y||
  4. ||x|| - ||y||
Question 2 Multiple Choice (Single Answer)

Which of the following is a property of a norm in a normed vector space?

  1. Positive Definiteness
  2. Homogeneity
  3. Triangle Inequality
  4. All of the above
Question 3 Multiple Choice (Single Answer)

A sequence {xn} in a normed vector space is called a Cauchy sequence if:

  1. ||xn - xm|| → 0 as n, m → ∞
  2. ||xn|| → 0 as n → ∞
  3. ||xn - xn+1|| → 0 as n → ∞
  4. ||xn + xn+1|| → 0 as n → ∞
Question 4 Multiple Choice (Single Answer)

A normed vector space that is also complete (i.e., every Cauchy sequence converges) is called a:

  1. Banach Space
  2. Hilbert Space
  3. Inner Product Space
  4. Metric Space
Question 5 Multiple Choice (Single Answer)

The norm of a linear operator T : X → Y between two normed vector spaces X and Y is defined as:

  1. ||T|| = sup{||Tx|| : x ∈ X, ||x|| ≤ 1}
  2. ||T|| = sup{||Tx|| : x ∈ X}
  3. ||T|| = inf{||Tx|| : x ∈ X, ||x|| = 1}
  4. ||T|| = inf{||Tx|| : x ∈ X}
Question 6 Multiple Choice (Single Answer)

In a normed vector space, the closed ball centered at x0 with radius r is defined as:

  1. {x ∈ X : ||x - x0|| ≤ r}
  2. {x ∈ X : ||x - x0|| < r}
  3. {x ∈ X : ||x - x0|| = r}
  4. {x ∈ X : ||x - x0|| > r}
Question 7 Multiple Choice (Single Answer)

Which of the following is a property of a closed ball in a normed vector space?

  1. It is a closed set.
  2. It is a bounded set.
  3. It is a convex set.
  4. All of the above
Question 8 Multiple Choice (Single Answer)

The dual space of a normed vector space X is denoted as:

  1. X*
  2. X+
  3. X-
  4. X0
Question 9 Multiple Choice (Single Answer)

In a normed vector space, the Hahn-Banach theorem states that:

  1. Every linear functional on a subspace can be extended to a linear functional on the whole space.
  2. Every bounded linear functional on a subspace can be extended to a bounded linear functional on the whole space.
  3. Every closed subspace of a normed vector space is complete.
  4. Every Cauchy sequence in a normed vector space converges.
Question 10 Multiple Choice (Single Answer)

Which of the following is an example of a Banach space?

  1. The space of continuous functions on a closed interval [a, b]
  2. The space of square-integrable functions on the real line
  3. The space of polynomials with real coefficients
  4. The space of sequences of real numbers
Question 11 Multiple Choice (Single Answer)

In a normed vector space, the open ball centered at x0 with radius r is defined as:

  1. {x ∈ X : ||x - x0|| < r}
  2. {x ∈ X : ||x - x0|| ≤ r}
  3. {x ∈ X : ||x - x0|| = r}
  4. {x ∈ X : ||x - x0|| > r}
Question 12 Multiple Choice (Single Answer)

Which of the following is a property of an open ball in a normed vector space?

  1. It is an open set.
  2. It is a bounded set.
  3. It is a convex set.
  4. All of the above
Question 13 Multiple Choice (Single Answer)

In a normed vector space, the unit ball is defined as:

  1. {x ∈ X : ||x|| ≤ 1}
  2. {x ∈ X : ||x|| < 1}
  3. {x ∈ X : ||x|| = 1}
  4. {x ∈ X : ||x|| > 1}
Question 14 Multiple Choice (Single Answer)

Which of the following is a property of the unit ball in a normed vector space?

  1. It is a closed set.
  2. It is a bounded set.
  3. It is a convex set.
  4. All of the above
Question 15 Multiple Choice (Single Answer)

In a normed vector space, the norm of a vector x is denoted as:

  1. ||x||
  2. ||x||
  3. |x|
  4. |x|
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