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Banach Spaces

Description: This quiz covers fundamental concepts and theorems related to Banach spaces, a vital topic in functional analysis.
Number of Questions: 15
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Tags: banach spaces functional analysis normed spaces completeness
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Let $X$ be a Banach space and $T: X \rightarrow X$ be a linear operator. Which of the following statements is true?

  1. If $T$ is bounded, then it is continuous.

  2. If $T$ is continuous, then it is bounded.

  3. If $T$ is compact, then it is bounded.

  4. If $T$ is bounded, then it is compact.

Show answer
Correct Option: A
Explanation:

In a Banach space, boundedness implies continuity due to the completeness of the space.

What is the Hahn-Banach theorem used for?

  1. Extending linear functionals on normed spaces.

  2. Finding the dual space of a Banach space.

  3. Characterizing bounded linear operators.

  4. All of the above.

Show answer
Correct Option: D
Explanation:

The Hahn-Banach theorem is a fundamental result in functional analysis with various applications, including extending linear functionals, finding dual spaces, and characterizing bounded linear operators.

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

  1. The space of continuous functions on a closed interval.

  2. The space of square-integrable functions on a measure space.

  3. The space of all polynomials with real coefficients.

  4. The space of all sequences of real numbers.

Show answer
Correct Option: A
Explanation:

The space of continuous functions on a closed interval is a Banach space under the supremum norm.

What is the Banach-Steinhaus theorem?

  1. A theorem about the boundedness of a family of linear operators.

  2. A theorem about the completeness of a Banach space.

  3. A theorem about the existence of a fixed point for a contraction mapping.

  4. A theorem about the existence of a dual space for a Banach space.

Show answer
Correct Option: A
Explanation:

The Banach-Steinhaus theorem states that if $X$ and $Y$ are Banach spaces and $T_n: X \rightarrow Y$ is a sequence of bounded linear operators such that $\sup_n |T_n| < \infty$, then the set ${T_n}$ is equicontinuous and pointwise bounded.

What is the Open Mapping Theorem?

  1. A theorem stating that a continuous bijective linear operator between Banach spaces is an open map.

  2. A theorem stating that a continuous linear operator between Banach spaces is closed.

  3. A theorem stating that a bounded linear operator between Banach spaces is compact.

  4. A theorem stating that a compact linear operator between Banach spaces is bounded.

Show answer
Correct Option: A
Explanation:

The Open Mapping Theorem states that if $X$ and $Y$ are Banach spaces and $T: X \rightarrow Y$ is a continuous bijective linear operator, then $T$ is an open map.

Which of the following is not a property of a Banach space?

  1. Completeness.

  2. Linearity.

  3. Normed.

  4. Separability.

Show answer
Correct Option: B
Explanation:

Banach spaces are normed and complete, but they are not necessarily linear.

What is the dual space of a Banach space?

  1. The space of all continuous linear functionals on the Banach space.

  2. The space of all bounded linear operators on the Banach space.

  3. The space of all compact linear operators on the Banach space.

  4. The space of all closed linear operators on the Banach space.

Show answer
Correct Option: A
Explanation:

The dual space of a Banach space $X$ is the space of all continuous linear functionals on $X$, denoted by $X^*$

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

  1. The space of continuous functions on a closed interval with the supremum norm and pointwise multiplication.

  2. The space of square-integrable functions on a measure space with the $L^2$ norm and pointwise multiplication.

  3. The space of all polynomials with real coefficients with the supremum norm and pointwise multiplication.

  4. The space of all sequences of real numbers with the supremum norm and pointwise multiplication.

Show answer
Correct Option: A
Explanation:

The space of continuous functions on a closed interval with the supremum norm and pointwise multiplication is a Banach algebra.

What is the Uniform Boundedness Principle?

  1. A theorem stating that if a family of linear operators between Banach spaces is pointwise bounded, then it is uniformly bounded.

  2. A theorem stating that if a family of linear operators between Banach spaces is equicontinuous, then it is uniformly bounded.

  3. A theorem stating that if a family of linear operators between Banach spaces is bounded, then it is equicontinuous.

  4. A theorem stating that if a family of linear operators between Banach spaces is compact, then it is bounded.

Show answer
Correct Option: A
Explanation:

The Uniform Boundedness Principle states that if $X$ and $Y$ are Banach spaces and ${T_n: X \rightarrow Y}$ is a family of linear operators such that $\sup_{x \in X} |T_n(x)| < \infty$, then $\sup_{n \in \mathbb{N}} |T_n| < \infty$.

Which of the following is a property of a reflexive Banach space?

  1. Every continuous linear functional on the space is bounded.

  2. Every bounded linear functional on the space is continuous.

  3. The space is isomorphic to its dual space.

  4. All of the above.

Show answer
Correct Option: D
Explanation:

A reflexive Banach space is a Banach space that is isomorphic to its dual space. Every continuous linear functional on a reflexive Banach space is bounded, and every bounded linear functional is continuous.

What is the Closed Graph Theorem?

  1. A theorem stating that if a linear operator between Banach spaces is closed, then its graph is closed.

  2. A theorem stating that if a linear operator between Banach spaces is bounded, then its graph is closed.

  3. A theorem stating that if a linear operator between Banach spaces is continuous, then its graph is closed.

  4. A theorem stating that if a linear operator between Banach spaces is compact, then its graph is closed.

Show answer
Correct Option: A
Explanation:

The Closed Graph Theorem states that if $X$ and $Y$ are Banach spaces and $T: X \rightarrow Y$ is a closed linear operator, then the graph of $T$, ${(x, T(x)) \in X \times Y}$, is a closed subset of $X \times Y$.

Which of the following is an example of a non-reflexive Banach space?

  1. The space of continuous functions on a closed interval.

  2. The space of square-integrable functions on a measure space.

  3. The space of all polynomials with real coefficients.

  4. The space of all sequences of real numbers.

Show answer
Correct Option: A
Explanation:

The space of continuous functions on a closed interval is an example of a non-reflexive Banach space.

What is the Principle of Uniform Boundedness?

  1. A theorem stating that if a family of linear operators between Banach spaces is pointwise bounded, then it is uniformly bounded.

  2. A theorem stating that if a family of linear operators between Banach spaces is equicontinuous, then it is uniformly bounded.

  3. A theorem stating that if a family of linear operators between Banach spaces is bounded, then it is equicontinuous.

  4. A theorem stating that if a family of linear operators between Banach spaces is compact, then it is bounded.

Show answer
Correct Option: A
Explanation:

The Principle of Uniform Boundedness states that if $X$ and $Y$ are Banach spaces and ${T_n: X \rightarrow Y}$ is a family of linear operators such that $\sup_{x \in X} |T_n(x)| < \infty$ for each $n \in \mathbb{N}$, then $\sup_{n \in \mathbb{N}} |T_n| < \infty$.

Which of the following is a property of a Banach space with a Schauder basis?

  1. Every element in the space can be represented as a unique infinite linear combination of the basis vectors.

  2. The space is separable.

  3. The space is reflexive.

  4. All of the above.

Show answer
Correct Option: D
Explanation:

A Banach space with a Schauder basis has the following properties: every element in the space can be represented as a unique infinite linear combination of the basis vectors, the space is separable, and the space is reflexive.

What is the Banach-Alaoglu theorem?

  1. A theorem stating that the closed unit ball in the dual space of a Banach space is weak*-compact.

  2. A theorem stating that the closed unit ball in a Banach space is weak-compact.

  3. A theorem stating that the closed unit ball in the dual space of a Banach space is norm-compact.

  4. A theorem stating that the closed unit ball in a Banach space is norm-compact.

Show answer
Correct Option: A
Explanation:

The Banach-Alaoglu theorem states that the closed unit ball in the dual space of a Banach space is weak*-compact.

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