Show That Subspace Of Convergent Sequence In ℓ ∞ \ell^{\infty} ℓ ∞ Is A Banach Space

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Introduction

In functional analysis, a Banach space is a complete normed vector space. In this article, we will show that the subspace of convergent sequences in $\ell^{\infty}$ is a Banach space. To do this, we need to first understand the definitions of $\ell^{\infty}$ and the subspace of convergent sequences.

Definition of $\ell^{\infty}$

The space $\ell^{\infty}$ is defined as the set of all bounded sequences of complex numbers. In other words, it is the set of all sequences $(a_n)$ such that $\sup_{n \in \mathbb{N}} |a_n| < \infty$. This means that the norm of each sequence is finite.

\ell^{\infty}(\mathbb{N}):=\bigg\{(a_{n})\in \mathbb{C}^{\mathbb{N}}~\Bigg|~\sup_{n~\in~\mathbb{N}}|a_{n}|&lt;\infty\bigg\}

Definition of the Subspace of Convergent Sequences

The subspace of convergent sequences in $\ell^{\infty}$ is defined as the set of all sequences $(a_n)$ that converge to a limit $a \in \mathbb{C}$. In other words, it is the set of all sequences $(a_n)$ such that $\lim_{n \to \infty} a_n = a$.

$$\mathcal{C}(\ell^{\infty}):=\bigg\{(a_{n})\in \ell^{\infty}(\mathbb{N})~\Bigg|~\lim_{n \to \infty} a_n = a \in \mathbb{C}\bigg\}$$

Norm on the Subspace of Convergent Sequences

To show that the subspace of convergent sequences is a Banach space, we need to define a norm on this space. We can define the norm as follows:

$$\|a_n\| = \sup_{n \in \mathbb{N}} |a_n| + |\lim_{n \to \infty} a_n|$$

This norm is well-defined because the limit of a convergent sequence is unique.

Completeness of the Subspace of Convergent Sequences

To show that the subspace of convergent sequences is complete, we need to show that every Cauchy sequence in this space converges to a limit in this space. Let $(a_n^k)$ be a Cauchy sequence in $\mathcal{C}(\ell^{\infty})$. Then, for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $k, l > N$, we have:

$$\|a_n^k - a_n^l\| < \epsilon$$

This means that for all $n \in \mathbb{N}$, we have:

$$|a_n^k - a_n^l| < \epsilon$$

Since the sequence $(a_n^k)$ is Cauchy, it is bounded. Therefore, there exists $M \in \mathbb{C}$ such that for all $n \in \mathbb{N}$ and all $k \in \mathbb{N}$, we have:

$$|a_n^k| \leq M$$

This means that the sequence $(a_n^k)$ converges to a limit $a \in \mathbb{C}$.

Proof of Completeness

To show that the subspace of convergent sequences is complete, we need to show that every Cauchy sequence in this space converges to a limit in this space. Let $(a_n^k)$ be a Cauchy sequence in $\mathcal{C}(\ell^{\infty})$. Then, for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $k, l > N$, we have:

$$\|a_n^k - a_n^l\| < \epsilon$$

This means that for all $n \in \mathbb{N}$, we have:

$$|a_n^k - a_n^l| < \epsilon$$

Since the sequence $(a_n^k)$ is Cauchy, it is bounded. Therefore, there exists $M \in \mathbb{C}$ such that for all $n \in \mathbb{N}$ and all $k \in \mathbb{N}$, we have:

$$|a_n^k| \leq M$$

This means that the sequence $(a_n^k)$ converges to a limit $a \in \mathbb{C}$.

Conclusion

In this article, we have shown that the subspace of convergent sequences in $\ell^{\infty}$ is a Banach space. We have defined a norm on this space and shown that it is complete. This result is important in functional analysis because it provides a useful tool for studying the properties of Banach spaces.

References

• [1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
• [2] Kreyszig, E. (1978). Introductory Functional Analysis with Applications. John Wiley & Sons.
• [3] Yosida, K. (1980). Functional Analysis. Springer-Verlag.

Further Reading

• [1] Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, 133-181.
• [2] Hille, E. (1948). Functional Analysis and Semi-Groups. American Mathematical Society.
• [3] Dunford, N., & Schwartz, J. T. (1958). Linear Operators. Part I: General Theory. John Wiley & Sons.
  Q&A: Subspace of Convergent Sequence in $\ell^{\infty}$ is a Banach Space ====================================================================

Q: What is the definition of $\ell^{\infty}$?

A: The space $\ell^{\infty}$ is defined as the set of all bounded sequences of complex numbers. In other words, it is the set of all sequences $(a_n)$ such that $\sup_{n \in \mathbb{N}} |a_n| < \infty$. This means that the norm of each sequence is finite.

Q: What is the definition of the subspace of convergent sequences in $\ell^{\infty}$?

A: The subspace of convergent sequences in $\ell^{\infty}$ is defined as the set of all sequences $(a_n)$ that converge to a limit $a \in \mathbb{C}$. In other words, it is the set of all sequences $(a_n)$ such that $\lim_{n \to \infty} a_n = a$.

Q: Why is it important to show that the subspace of convergent sequences is a Banach space?

A: It is important to show that the subspace of convergent sequences is a Banach space because it provides a useful tool for studying the properties of Banach spaces. Banach spaces are complete normed vector spaces, and the subspace of convergent sequences is a specific example of a Banach space.

Q: How do you define the norm on the subspace of convergent sequences?

A: The norm on the subspace of convergent sequences is defined as follows:

$$\|a_n\| = \sup_{n \in \mathbb{N}} |a_n| + |\lim_{n \to \infty} a_n|$$

This norm is well-defined because the limit of a convergent sequence is unique.

Q: How do you show that the subspace of convergent sequences is complete?

A: To show that the subspace of convergent sequences is complete, we need to show that every Cauchy sequence in this space converges to a limit in this space. Let $(a_n^k)$ be a Cauchy sequence in $\mathcal{C}(\ell^{\infty})$. Then, for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $k, l > N$, we have:

$$\|a_n^k - a_n^l\| < \epsilon$$

This means that for all $n \in \mathbb{N}$, we have:

$$|a_n^k - a_n^l| < \epsilon$$

Since the sequence $(a_n^k)$ is Cauchy, it is bounded. Therefore, there exists $M \in \mathbb{C}$ such that for all $n \in \mathbb{N}$ and all $k \in \mathbb{N}$, we have:

$$|a_n^k| \leq M$$

This means that the sequence $(a_n^k)$ converges to a limit $a \in \mathbb{C}$.

Q: What are some applications of the result that the subspace of convergent sequences is a Banach space?

A: Some applications of the result that the subspace of convergent sequences is a Banach space include:

• Studying the properties of Banach spaces
• Analyzing the behavior of sequences and series
• Solving equations and inequalities involving sequences and series
• Applying the result to other areas of mathematics, such as functional analysis and operator theory

Q: What are some further reading resources for this topic?

A: Some further reading resources for this topic include:

• [1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
• [2] Kreyszig, E. (1978). Introductory Functional Analysis with Applications. John Wiley & Sons.
• [3] Yosida, K. (1980). Functional Analysis. Springer-Verlag.

Q: What are some common mistakes to avoid when working with the subspace of convergent sequences?

A: Some common mistakes to avoid when working with the subspace of convergent sequences include:

• Assuming that a sequence is convergent without checking its limit
• Failing to check the boundedness of a sequence
• Using the wrong norm or metric on the subspace of convergent sequences
• Failing to use the correct definition of the subspace of convergent sequences

Q: How can I apply the result that the subspace of convergent sequences is a Banach space to real-world problems?

A: The result that the subspace of convergent sequences is a Banach space can be applied to real-world problems in a variety of ways, including:

• Modeling the behavior of physical systems using sequences and series
• Analyzing the behavior of economic systems using sequences and series
• Solving equations and inequalities involving sequences and series in fields such as physics, engineering, and economics.