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Normed Vector Spaces

Description: This quiz covers the fundamental concepts and properties of normed vector spaces, a cornerstone of functional analysis and linear algebra.
Number of Questions: 15
Created by:
Tags: normed vector spaces vector norms cauchy sequences completeness banach spaces
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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||

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Correct Option: A
Explanation:

The distance between two vectors in a normed vector space is given by the norm of their difference, which is denoted as ||x - y||.

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

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Correct Option: D
Explanation:

A norm in a normed vector space satisfies three fundamental properties: positive definiteness (||x|| ≥ 0 for all x), homogeneity (||αx|| = |α| ||x|| for all scalars α and vectors x), and the triangle inequality (||x + y|| ≤ ||x|| + ||y|| for all vectors x and y).

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 → ∞

Show answer
Correct Option: A
Explanation:

A sequence {xn} in a normed vector space is a Cauchy sequence if for any ε > 0, there exists an integer N such that ||xn - xm|| < ε for all n, m ≥ N.

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

Show answer
Correct Option: A
Explanation:

A normed vector space that is complete is called a Banach space. Banach spaces are named after the Polish mathematician Stefan Banach, who made significant contributions to the study of functional analysis.

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}

Show answer
Correct Option: A
Explanation:

The norm of a linear operator T : X → Y is defined as the least upper bound of the norms of Tx for all unit vectors x in X.

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}

Show answer
Correct Option: A
Explanation:

The closed ball centered at x0 with radius r is the set of all vectors x in the normed vector space such that the distance between x and x0 is less than or equal to r.

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

Show answer
Correct Option: D
Explanation:

A closed ball in a normed vector space is a closed, bounded, and convex set.

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

  1. X*

  2. X+

  3. X-

  4. X0

Show answer
Correct Option: A
Explanation:

The dual space of a normed vector space X is the space of all bounded linear functionals on X, denoted as X*.

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.

Show answer
Correct Option: A
Explanation:

The Hahn-Banach theorem is a fundamental result in functional analysis that guarantees the existence of linear extensions of linear functionals defined on subspaces of a normed vector space.

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

Show answer
Correct Option: A
Explanation:

The space of continuous functions on a closed interval [a, b] with the supremum norm is a Banach space, known as the Banach space of continuous functions.

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}

Show answer
Correct Option: A
Explanation:

The open ball centered at x0 with radius r is the set of all vectors x in the normed vector space such that the distance between x and x0 is less than r.

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

Show answer
Correct Option: A
Explanation:

An open ball in a normed vector space is an open set.

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}

Show answer
Correct Option: A
Explanation:

The unit ball in a normed vector space is the set of all vectors x such that the norm of x is less than or equal to 1.

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

Show answer
Correct Option: D
Explanation:

The unit ball in a normed vector space is a closed, bounded, and convex set.

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

  1. ||x||

  2. ||x||

  3. |x|

  4. |x|

Show answer
Correct Option: A
Explanation:

The norm of a vector x in a normed vector space is denoted by the double vertical bars ||x||.

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