Definition of Convergent Sequences

Oct 13, 2025 by pinkandtrash.com 35 views

Introduction

In this article, we will explore the concept of a Banach space and show that the subspace of convergent sequences in $\ell^{\infty}$ is indeed a Banach space. We will begin by defining the space $\ell^{\infty}$ and its subspace of convergent sequences, and then proceed to prove that this subspace is complete and thus a Banach space.

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 the supremum of the absolute values of the terms is finite. Mathematically, this can be expressed as:

\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 Convergent Sequences

A sequence $(a_n)$ is said to be convergent if there exists a complex number $a$ such that for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $n \geq N$ , $|a_n - a| < \epsilon$ . The number $a$ is called the limit of the sequence.

Subspace of Convergent Sequences

The subspace of convergent sequences in $\ell^{\infty}$ is denoted by $\mathcal{C}(\ell^{\infty})$ . It is defined as the set of all sequences in $\ell^{\infty}$ that are convergent. In other words:

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

Proof that $\mathcal{C}(\ell^{\infty})$ is a Banach Space

To show that $\mathcal{C}(\ell^{\infty})$ is a Banach space, we need to prove that it is complete. In other words, we need to show that every Cauchy sequence in $\mathcal{C}(\ell^{\infty})$ converges to an element in $\mathcal{C}(\ell^{\infty})$ .

Let $(a_n)$ be a Cauchy sequence in $\mathcal{C}(\ell^{\infty})$ . Then, for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $m, n \geq N$ , $|a_m - a_n| < \epsilon$ . This implies that the sequence $(a_n)$ is a Cauchy sequence in $\ell^{\infty}$ .

Since $\ell^{\infty}$ is a Banach space, the sequence $(a_n)$ converges to an element $a \in \ell^{\infty}$ . We need to show that $a \in \mathcal{C}(\ell^{\infty})$ . In other words, we need to show that there exists a complex number $b$ such that for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $n \geq N$ , $|a_n - b| < \epsilon$ .

Let $b$ be the limit of the sequence $(a_n)$ in $\ell^{\infty}$ . Then, for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $n \geq N$ , $|a_n - b| < \epsilon$ . This implies that $a = b$ , and therefore, $a \in \mathcal{C}(\ell^{\infty})$ .

Conclusion

In this article, we have shown that the subspace of convergent sequences in $\ell^{\infty}$ is a Banach space. We have defined the space $\ell^{\infty}$ and its subspace of convergent sequences, and then proceeded to prove that this subspace is complete and thus a Banach space.

References

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

Q: What is the definition of a Banach space?

A: A Banach space is a complete normed vector space. In other words, it is a vector space that is complete with respect to the norm.

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

A: The space $\ell^{\infty}$ is the set of all bounded sequences of complex numbers. In other words, it is the set of all sequences $(a_n)$ such that the supremum of the absolute values of the terms is finite.

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

A: The subspace of convergent sequences in $\ell^{\infty}$ is denoted by $\mathcal{C}(\ell^{\infty})$ . It is defined as the set of all sequences in $\ell^{\infty}$ that are convergent.

Q: Why is it important to show that $\mathcal{C}(\ell^{\infty})$ is a Banach space?

A: It is important to show that $\mathcal{C}(\ell^{\infty})$ is a Banach space because it has many applications in functional analysis and operator theory. For example, it is used to study the properties of linear operators on Banach spaces.

Q: How do you prove that a space is a Banach space?

A: To prove that a space is a Banach space, you need to show that it is complete. In other words, you need to show that every Cauchy sequence in the space converges to an element in the space.

Q: What is a Cauchy sequence?

A: A Cauchy sequence is a sequence of elements in a metric space such that for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $m, n \geq N$ , the distance between the $m$ th and $n$ th elements is less than $\epsilon$ .

Q: How do you show that a Cauchy sequence converges to an element in a space?

A: To show that a Cauchy sequence converges to an element in a space, you need to show that the sequence converges to a limit in the space. In other words, you need to show that there exists an element in the space such that for every $\epsilon > 0$ , there exists a natural number $N$ such that for all $n \geq N$ , the distance between the $n$ th element and the limit is less than $\epsilon$ .

Q: What is the significance of the result that $\mathcal{C}(\ell^{\infty})$ is a Banach space?

A: The result that $\mathcal{C}(\ell^{\infty})$ is a Banach space has many significant implications in functional analysis and operator theory. For example, it is used to study the properties of linear operators on Banach spaces and to prove the existence of certain types of operators.

Q: What are some of the applications of the result that $\mathcal{C}(\ell^{\infty})$ is a Banach space?

A: Some of the applications of the result that $\mathcal{C}(\ell^{\infty})$ is a Banach space include:

Studying the properties of linear operators on Banach spaces
Proving the existence of certain types of operators
Developing new techniques for solving problems in functional analysis and operator theory

Q: What are some of the challenges associated with proving that a space is a Banach space?

A: Some of the challenges associated with proving that a space is a Banach space include:

Showing that the space is complete
Showing that the space has a well-defined norm
Overcoming technical difficulties in the proof

Q: What are some of the tools and techniques used to prove that a space is a Banach space?

A: Some of the tools and techniques used to prove that a space is a Banach space include:

Using the definition of a Banach space
Applying the Cauchy criterion for convergence
Using the properties of the norm on the space
Developing new techniques for solving problems in functional analysis and operator theory