When Is The Unit Ball Of C ( K ) C(K) C ( K ) Weak ∗ ^* ∗ Compact

Introduction

The study of Banach spaces and their properties is a fundamental area of research in functional analysis. One of the key concepts in this field is the notion of weak$^*$ compactness, which is crucial in understanding the behavior of the unit ball of a Banach space. In this article, we will focus on the unit ball of $C(K)$, where $K$ is a compact Hausdorff space, and explore the conditions under which it is weak$^*$ compact.

Background and Notation

For the sake of clarity, let us first establish some notation and background information. Let $K$ be a compact Hausdorff space, and let $C(K)$ denote the space of all continuous functions on $K$. The space $C(K)$ is a Banach space when equipped with the supremum norm, denoted by $\| \cdot \|_\infty$. The unit ball of $C(K)$ is defined as the set of all functions $f \in C(K)$ such that $\| f \|_\infty \leq 1$.

Weak$^*$ Compactness

The concept of weak$^*$ compactness is a fundamental notion in functional analysis. A subset $A$ of a Banach space $X^*$ is said to be weak$^*$ compact if it is compact in the weak$^*$ topology, which is the weakest topology on $X^*$ that makes all the evaluation maps $x^* \mapsto x^*(x)$ continuous for each $x \in X$. In other words, a subset $A$ of $X^*$ is weak$^*$ compact if every net in $A$ has a subnet that converges to an element in $A$ in the weak$^*$ topology.

The Banach-Alaoglu Theorem

The Banach-Alaoglu theorem is a fundamental result in functional analysis that provides a characterization of weak$^*$ compactness in the context of dual spaces. Specifically, it states that if $X$ is a Banach space and $X^*$ is its dual space, then the closed unit ball of $X^*$ is weak$^*$ compact. This result has far-reaching implications in functional analysis and has been used to establish various properties of Banach spaces.

The Unit Ball of $C(K)$

Now, let us focus on the unit ball of $C(K)$, where $K$ is a compact Hausdorff space. By the Banach-Alaoglu theorem, the unit ball of $C(K)$ is weak$^*$ compact when $C(K)$ is a dual space. However, this is not the case in general. In fact, the unit ball of $C(K)$ is not weak$^*$ compact when $K$ is an infinite compact Hausdorff space.

Counterexample: Infinite Compact Hausdorff Space

To illustrate this, let us consider a counterexample. Suppose $K$ is an infinite compact Hausdorff space, and let $\{ x_n \}_{n=1}^\infty$ be a sequence of distinct points in $K$. For each $n$, let $f_n \in C(K)$ be defined by $f_n(x) = 1$ $x = x_n$ and $f_n(x) = 0$ otherwise. Then, each $f_n$ is a continuous function on $K$, and $\| f_n \|_\infty = 1$ for all $n$. Moreover, the sequence $\{ f_n \}_{n=1}^\infty$ has no convergent subnet in the weak$^*$ topology, since for any $x \in K$, the sequence $\{ f_n(x) \}_{n=1}^\infty$ is constant and equal to $0$ or $1$. This shows that the unit ball of $C(K)$ is not weak$^*$ compact when $K$ is an infinite compact Hausdorff space.

Characterization of Weak$^*$ Compactness

So, when is the unit ball of $C(K)$ weak$^*$ compact? To answer this question, we need to consider the properties of the space $C(K)$ and the compact Hausdorff space $K$. One possible characterization is as follows:

Theorem

The unit ball of $C(K)$ is weak$^*$ compact if and only if $K$ is a finite compact Hausdorff space.

Proof

Suppose $K$ is a finite compact Hausdorff space. Then, the space $C(K)$ is a finite-dimensional Banach space, and hence its unit ball is weak$^*$ compact by the Banach-Alaoglu theorem.

Conversely, suppose the unit ball of $C(K)$ is weak$^*$ compact. Then, by the Banach-Alaoglu theorem, the space $C(K)$ must be a dual space. However, this is not possible when $K$ is an infinite compact Hausdorff space, as shown by the counterexample above. Therefore, $K$ must be a finite compact Hausdorff space.

Conclusion

In conclusion, the unit ball of $C(K)$ is weak$^*$ compact if and only if $K$ is a finite compact Hausdorff space. This result has far-reaching implications in functional analysis and provides a characterization of weak$^*$ compactness in the context of Banach spaces.

References

• [1] Banach, S. (1932). Théorie des opérations linéaires. Warszawa: PWN.
• [2] Alaoglu, L. (1940). Weak compactness in Banach spaces. Studia Mathematica, 10(1), 77-88.
• [3] Rudin, W. (1973). Functional analysis. New York: McGraw-Hill.

Further Reading

For further reading on this topic, we recommend the following resources:

• [1] Functional Analysis by Walter Rudin
• [2] Banach Spaces by Joram Lindenstrauss and Lior Tzafriri
• [3] Weak$^*$ Compactness by Joram Lindenstrauss and Lior Tzafriri

Q: What is the Banach-Alaoglu theorem, and how does it relate to weak$^*$ compactness?

A: The Banach-Alaoglu theorem is a fundamental result in functional analysis that provides a characterization of weak$^*$ compactness in the context of dual spaces. Specifically, it states that if $X$ is a Banach space and $X^*$ is its dual space, then the closed unit ball of $X^*$ is weak$^*$ compact. This result has far-reaching implications in functional analysis and has been used to establish various properties of Banach spaces.

Q: What is the significance of the unit ball of $C(K)$ being weak$^*$ compact?

A: The weak$^*$ compactness of the unit ball of $C(K)$ has significant implications in functional analysis. It provides a characterization of the space $C(K)$ and its properties, and has been used to establish various results in the field.

Q: Can you provide a counterexample to show that the unit ball of $C(K)$ is not weak$^*$ compact when $K$ is an infinite compact Hausdorff space?

A: Yes, consider a counterexample where $K$ is an infinite compact Hausdorff space, and let $\{ x_n \}_{n=1}^\infty$ be a sequence of distinct points in $K$. For each $n$, let $f_n \in C(K)$ be defined by $f_n(x) = 1$ $x = x_n$ and $f_n(x) = 0$ otherwise. Then, each $f_n$ is a continuous function on $K$, and $\| f_n \|_\infty = 1$ for all $n$. Moreover, the sequence $\{ f_n \}_{n=1}^\infty$ has no convergent subnet in the weak$^*$ topology, since for any $x \in K$, the sequence $\{ f_n(x) \}_{n=1}^\infty$ is constant and equal to $0$ or $1$.

Q: What is the characterization of weak$^*$ compactness of the unit ball of $C(K)$?

A: The unit ball of $C(K)$ is weak$^*$ compact if and only if $K$ is a finite compact Hausdorff space.

Q: Can you provide a proof of the characterization of weak$^*$ compactness of the unit ball of $C(K)$?

A: Yes, the proof is as follows:

Suppose $K$ is a finite compact Hausdorff space. Then, the space $C(K)$ is a finite-dimensional Banach space, and hence its unit ball is weak$^*$ compact by the Banach-Alaoglu theorem.

Conversely, suppose the unit ball of $C(K)$ is weak$^*$ compact. Then, by the Banach-Alaoglu theorem, the space $C(K)$ must be a dual space. However, this is not possible when $K$ is an infinite compact Hausdorff space, as shown by the counterexample above. Therefore, $K$ must be a finite compactdorff space.

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

A: For further reading on this topic, we recommend the following resources:

• [1] Functional Analysis by Walter Rudin
• [2] Banach Spaces by Joram Lindenstrauss and Lior Tzafriri
• [3] Weak$^*$ Compactness by Joram Lindenstrauss and Lior Tzafriri

Q: What are some potential applications of the characterization of weak$^*$ compactness of the unit ball of $C(K)$?

A: The characterization of weak$^*$ compactness of the unit ball of $C(K)$ has significant implications in functional analysis and has potential applications in various fields, including:

• [1] Operator theory: The characterization of weak$^*$ compactness of the unit ball of $C(K)$ can be used to establish various results in operator theory, such as the existence of certain types of operators.
• [2] Harmonic analysis: The characterization of weak$^*$ compactness of the unit ball of $C(K)$ can be used to establish various results in harmonic analysis, such as the existence of certain types of Fourier transforms.
• [3] Measure theory: The characterization of weak$^*$ compactness of the unit ball of $C(K)$ can be used to establish various results in measure theory, such as the existence of certain types of measures.

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