Separability of the Continuous Dual

Oct 13, 2025 by pinkandtrash.com 120 views

Introduction

In functional analysis, the concept of separability plays a crucial role in understanding the properties of normed spaces. A normed space is said to be separable if it contains a countable dense subset. The separability of a space has significant implications on its dual space, which is the space of all continuous linear functionals on the original space. In this article, we will explore the relationship between the separability of the continuous dual and the original space, and discuss a proof of the proposition that if the continuous dual of a normed space is separable, then the original space is also separable.

Background

Let $X$ be a normed space. The dual space $X^*$ of $X$ is the space of all continuous linear functionals on $X$ . A linear functional $f \in X^*$ is said to be continuous if there exists a constant $c > 0$ such that $\|f(x)\| \leq c\|x\|$ for all $x \in X$ . The norm of a linear functional $f \in X^*$ is defined as $\|f\| = \sup \{\|f(x)\| : x \in X, \|x\| \leq 1\}$ .

Separability of the Continuous Dual

The continuous dual space $X^*$ is said to be separable if it contains a countable dense subset. In other words, $X^*$ is separable if there exists a countable set $D \subset X^*$ such that for any $f \in X^*$ and any $\epsilon > 0$ , there exists $g \in D$ such that $\|f - g\| < \epsilon$ .

Proposition

Proposition. Let $X$ be a normed space. If $X^*$ is separable, then $X$ is separable.

Proof

To prove this proposition, we will use the following theorem:

Theorem. Let $X$ be a normed space and $Y$ be a subspace of $X$ . If $Y$ is separable, then $X/Y$ is separable.

We will also use the following lemma:

Lemma. Let $X$ be a normed space and $Y$ be a subspace of $X$ . If $X/Y$ is separable, then $X$ is separable.

Step 1: Constructing a Countable Dense Subset of $X$

Let $D \subset X^*$ be a countable dense subset of $X^*$ . For each $n \in \mathbb{N}$ , let $D_n = \{x \in X : |f(x)| \leq n \text{ for all } f \in D\}$ . We claim that $D_n$ is a closed subspace of $X$ .

To see this, let $x \in X$ and let $\{x_k\}$ be a sequence in $D_n$ such that $x_k \to x$ . For each $f \in D$ , we have $|f(x_k)| \leq n$ for all $k$ . Since $f$ is continuous, we have $f(x) = \lim_{k \to \infty} f(x_k)$ . Therefore, $|f(x)| \leq n$ for all $f \in D$ , and hence $x \in D_n$ .

Step 2: Showing that $D_n$ is Separable

We claim that $D_n$ is separable. To see this, let $D_n' = \{x \in D_n : \|x\| \leq n\}$ . We claim that $D_n'$ is a countable dense subset of $D_n$ .

To see this, let $x \in D_n$ and let $\epsilon > 0$ . Since $D$ is dense in $X^*$ , there exists $f \in D$ such that $\|f - x^*\| < \epsilon$ , where $x^* \in X^*$ is the functional defined by $x^*(y) = y(x)$ for all $y \in X$ . Since $f \in D$ , we have $|f(y)| \leq n$ for all $y \in D_n$ . Therefore, $\|x^*\| \leq n$ , and hence $x^* \in D_n'$ . Since $\|x - x^*\| = \|f - x^*\| < \epsilon$ , we have $x \in \overline{D_n'}$ , and hence $D_n'$ is dense in $D_n$ .

Step 3: Showing that $X$ is Separable

We claim that $X$ is separable. To see this, let $D = \bigcup_{n=1}^{\infty} D_n$ . We claim that $D$ is a countable dense subset of $X$ .

To see this, let $x \in X$ and let $\epsilon > 0$ . Since $D$ is dense in $X^*$ , there exists $f \in D$ such that $\|f - x^*\| < \epsilon$ , where $x^* \in X^*$ is the functional defined by $x^*(y) = y(x)$ for all $y \in X$ . Since $f \in D$ , there exists $n \in \mathbb{N}$ such that $f \in D_n$ . Therefore, $x \in D_n$ , and hence $x \in \overline{D}$ .

Conclusion

In this article, we have discussed the relationship between the separability of the continuous dual and the original space. We have proved that if the continuous dual of a normed space is separable, then the original space is also separable. This result has significant implications in functional analysis, and it has been used in various applications, including the study of Banach spaces and the theory of operator algebras.

References

[1] Albiac, F., & Kalton, N. J. (2006). Topics in Banach space theory. Springer.
[2] Dunford, N., & Schwartz, J. T. (1958). Linear operators. Part I: General theory. Wiley.
[3] Rudin, W. (1973). Functional analysis. McGraw-Hill.

Introduction

In our previous article, we discussed the relationship between the separability of the continuous dual and the original space. We proved that if the continuous dual of a normed space is separable, then the original space is also separable. In this article, we will answer some frequently asked questions about this topic.

Q: What is the significance of separability in functional analysis?

A: Separability is a fundamental concept in functional analysis, and it has significant implications on the properties of normed spaces. A separable space contains a countable dense subset, which means that every element in the space can be approximated by a sequence of elements from a countable set.

Q: What is the relationship between the separability of the continuous dual and the original space?

A: We have proved that if the continuous dual of a normed space is separable, then the original space is also separable. This result has significant implications in functional analysis, and it has been used in various applications, including the study of Banach spaces and the theory of operator algebras.

Q: How can we construct a countable dense subset of the original space?

A: To construct a countable dense subset of the original space, we can use the following steps:

Construct a countable dense subset of the continuous dual space.
Use the Hahn-Banach theorem to extend each element of the countable dense subset of the continuous dual space to an element of the original space.
Show that the set of extended elements is a countable dense subset of the original space.

Q: What are some applications of the separability of the continuous dual and the original space?

A: The separability of the continuous dual and the original space has significant implications in various areas of mathematics, including:

Banach space theory: The separability of the continuous dual and the original space is a fundamental concept in Banach space theory, and it has been used to study the properties of Banach spaces.
Operator algebra theory: The separability of the continuous dual and the original space is also a fundamental concept in operator algebra theory, and it has been used to study the properties of operator algebras.
Functional analysis: The separability of the continuous dual and the original space is a fundamental concept in functional analysis, and it has been used to study the properties of normed spaces.

Q: What are some open problems related to the separability of the continuous dual and the original space?

A: Some open problems related to the separability of the continuous dual and the original space include:

The separability of the continuous dual of a Banach space: It is not known whether the continuous dual of a Banach space is always separable.
The separability of the original space of a Banach space: It is not known whether the original space of a Banach space is always separable.
The relationship between the separability of the continuous dual and the original space: It is not known whether the separability of the continuous dual implies the separability of the original space in all cases.

Conclusion

In this article, we have answered some frequently asked questions about the separability of the continuous dual and the original space. We have discussed the significance of separability in functional analysis, the relationship between the separability of the continuous dual and the original space, and some applications of this result. We have also discussed some open problems related to this topic.

References

[1] Albiac, F., & Kalton, N. J. (2006). Topics in Banach space theory. Springer.
[2] Dunford, N., & Schwartz, J. T. (1958). Linear operators. Part I: General theory. Wiley.
[3] Rudin, W. (1973). Functional analysis. McGraw-Hill.