Separability of Dual Spaces

Oct 13, 2025 by windailys.com 96 views

Introduction

In functional analysis, a normed space is a vector space equipped with a norm, which is a way of measuring the size or length of vectors in the space. One of the fundamental properties of a normed space is its separability, which means that the space has a countable dense subset. In this article, we will discuss the separability of normed spaces and provide a proof of the proposition that if the dual space of a normed space is separable, then the original space is also separable.

Separable Spaces

A normed space $X$ is said to be separable if it has a countable dense subset, i.e., there exists a countable set $D \subset X$ such that for any $x \in X$ and any $\epsilon > 0$ , there exists $d \in D$ such that $\|x - d\| < \epsilon$ . In other words, the set $D$ is dense in $X$ .

Weak Topology

The weak topology on a normed space $X$ is the weakest topology on $X$ that makes all the linear functionals in $X^*$ continuous. A set $U \subset X$ is said to be weakly open if for any $x \in U$ , there exists a finite set of linear functionals $f_1, \ldots, f_n \in X^*$ and a neighborhood $V$ of $0$ in $\mathbb{R}^n$ such that $U \cap \{x \in X : f_i(x) \in V, 1 \leq i \leq n\} \neq \emptyset$ .

Separability of Dual Spaces

The dual space $X^*$ of a normed space $X$ is the set of all continuous linear functionals on $X$ . A normed space $X$ is said to be separable if its dual space $X^*$ is separable.

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 let $Y$ 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 let $Y$ be a closed subspace of $X$ . If $X/Y$ is separable, then $X$ is separable.

Step 1: Reduction to the Case of a Closed Subspace

Let $X$ be a normed space and suppose that $X^*$ is separable. We need to show that $X$ is separable. Let $D \subset X^*$ be a countable dense subset of $X^*$ . We will show that $X$ is separable by reducing the problem to the case of a closed subspace.

Let $Y$ be the closure of the span of $D$ in $X$ . Then $Y$ is a closed subspace of $X$ . We will show that $X/Y$ is separable.

Step 2: Separability of $X/Y$

Let $f_1, \ldots, f_n \in X^*$ be a finite set of linear functionals such that the set $U = \{x \in X : f_i(x) = 0, 1 \leq i \leq n\}$ is a weakly open neighborhood of $0$ in $X$ . We need to show that $X/Y$ is separable.

Let $V$ be a neighborhood of $0$ in $\mathbb{R}^n$ such that $U \cap \{x \in X : f_i(x) \in V, 1 \leq i \leq n\} \neq \emptyset$ . We will show that there exists a countable set $E \subset X/Y$ such that for any $x \in X/Y$ and any $\epsilon > 0$ , there exists $e \in E$ such that $\|x - e\| < \epsilon$ .

Let $x \in X/Y$ and let $\epsilon > 0$ . We need to find $e \in E$ such that $\|x - e\| < \epsilon$ . Let $y \in X$ be a representative of $x$ . Then $y \in U$ and $f_i(y) \in V$ for $1 \leq i \leq n$ .

Since $D$ is dense in $X^*$ , there exists $d \in D$ such that $\|f_i - d\| < \epsilon/n$ for $1 \leq i \leq n$ . Let $e \in X/Y$ be the equivalence class of $d(y)$ . Then $\|x - e\| < \epsilon$ .

Step 3: Separability of $X$

We have shown that $X/Y$ is separable. By the lemma, we conclude that $X$ is separable.

Conclusion

We have proved the proposition that if the dual space of a normed space is separable, then the original space is also separable. This result is a fundamental tool in functional analysis and has many applications in mathematics and physics.

References

[1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
[2] Conway, J. B. (1990). A Course in Functional Analysis. Springer-Verlag.
[3] Krein, M. G. (1950). Principles of Linear Analysis. Moscow: Fizmatgiz.

Additional Information

For the following proposition:

Proposition. $X$ normed space. If $X^*$ separable, then $X$ separable.

Introduction

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

Q: What is the difference between separability and completeness?

A: Separability and completeness are two distinct properties of normed spaces. A normed space is said to be separable if it has a countable dense subset, while a normed space is said to be complete if every Cauchy sequence in the space converges to a point in the space.

Q: Can a non-separable space be complete?

A: Yes, a non-separable space can be complete. For example, the space $l^{\infty}$ of all bounded sequences of real numbers is complete, but it is not separable.

Q: Can a separable space be non-complete?

A: Yes, a separable space can be non-complete. For example, the space $c_{0}$ of all sequences of real numbers that converge to zero is separable, but it is not complete.

Q: What is the relationship between separability and the weak topology?

A: The weak topology on a normed space is the weakest topology that makes all the linear functionals in the space continuous. A normed space is said to be separable if it has a countable dense subset with respect to the weak topology.

Q: Can a non-separable space have a separable dual?

A: Yes, a non-separable space can have a separable dual. For example, the space $l^{1}$ of all absolutely convergent sequences of real numbers is separable, but its dual space is not separable.

Q: Can a separable space have a non-separable dual?

A: Yes, a separable space can have a non-separable dual. For example, the space $l^{\infty}$ of all bounded sequences of real numbers is separable, but its dual space is not separable.

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

A: Separability is a fundamental concept in functional analysis, and it has many applications in mathematics and physics. For example, separability is used to study the properties of Banach spaces, Hilbert spaces, and other types of normed spaces.

Q: Can separability be used to study the properties of operators?

A: Yes, separability can be used to study the properties of operators. For example, separability is used to study the properties of compact operators, Fredholm operators, and other types of operators.

Q: What are some examples of separable and non-separable spaces?

A: Some examples of separable spaces include:

The space $l^{1}$ of all absolutely convergent sequences of real numbers
The space $c_{0}$ of all sequences of real numbers that converge to zero
The space $l^{2}$ of all square-summable sequences of real numbers

Some examples of non-separable spaces include:

The space $l^{\infty}$ of all bounded sequences of real numbers
The space $C[0,1]$ of all continuous functions on the interval $[0,1]$
The space $L^{p}[0,1]$ of all $p$ -integrable functions on the interval $[0,1]$ for $1 \leq p < \infty$

Conclusion

In this article, we have answered some frequently asked questions about separability of normed spaces. We have discussed the relationship between separability and completeness, the weak topology, and the properties of operators. We have also provided some examples of separable and non-separable spaces.