Show That T T T Is Bounded If And Only If V V V Is Closed.

Introduction

In functional analysis, a closed linear operator is a linear operator that maps a subspace of a Banach space to another Banach space, and its graph is a closed set. In this discussion, we will explore the relationship between a closed linear operator and the closedness of its domain. Specifically, we will show that a closed linear operator $T$ is bounded if and only if its domain $V$ is closed.

Preliminaries

Before we dive into the main proof, let's recall some definitions and properties.

• A Banach space is a complete normed vector space.
• A linear operator $T$ between two Banach spaces $X$ and $Y$ is said to be closed if its graph is a closed set in the product space $X \times Y$.
• A subspace $V$ of a Banach space $X$ is said to be closed if it contains all its limit points.

The "Only If" Part

Suppose $T$ is bounded. We need to show that $V$ is closed. Let $x \in \overline{V}$, where $\overline{V}$ denotes the closure of $V$. We need to show that $x \in V$.

Since $T$ is bounded, it is continuous. Therefore, for any sequence $\{x_n\} \subset V$ that converges to $x$, we have

$$\lim_{n \to \infty} T(x_n) = T(x).$$

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Since $T$ is closed, its graph is a closed set. Therefore, if $\{x_n\}$ is a sequence in $V$ that converges to $x$, then $\{T(x_n)\}$ is a sequence in $Y$ that converges to $T(x)$. Since $T$ is bounded, the sequence $\{T(x_n)\}$ is Cauchy. Therefore, since $Y$ is complete, the sequence $\{T(x_n)\}$ converges to some $y \in Y$.

Q: What is the relationship between a closed linear operator and the closedness of its domain?

A: A closed linear operator $T$ is bounded if and only if its domain $V$ is closed.

Q: Why is it important to show that $T$ is bounded if and only if $V$ is closed?

A: This result is important because it provides a necessary and sufficient condition for a closed linear operator to be bounded. This condition can be used to determine whether a closed linear operator is bounded or not.

Q: What is the "only if" part of the proof?

A: The "only if" part of the proof states that if $T$ is bounded, then $V$ is closed. This means that if a closed linear operator is bounded, then its domain is also closed.

Q: How do you show that $V$ is closed if $T$ is bounded?

A: To show that $V$ is closed if $T$ is bounded, we need to show that if $x \in \overline{V}$, then $x \in V$. We can do this by using the fact that $T$ is bounded and the definition of a closed set.

Q: What is the "if" part of the proof?

A: The "if" part of the proof states that if $V$ is closed, then $T$ is bounded. This means that if the domain of a closed linear operator is closed, then the operator is also bounded.

Q: How do you show that $T$ is bounded if $V$ is closed?

A: To show that $T$ is bounded if $V$ is closed, we need to show that $T$ is continuous. We can do this by using the fact that $V$ is closed and the definition of a continuous function.

Q: What are some common applications of this result?

A: This result has many applications in functional analysis, particularly in the study of closed linear operators and their properties. Some common applications include:

• The study of closed linear operators on Banach spaces
• The study of the properties of closed linear operators, such as their boundedness and continuity
• The study of the relationship between closed linear operators and their domains

Q: What are some common mistakes to avoid when proving this result?

A: Some common mistakes to avoid when proving this result include:

• Assuming that $T$ is bounded without showing that $V$ is closed
• Assuming that $V$ is closed without showing that $T$ is bounded
• Failing to use the definition of a closed set and a continuous function correctly

Q: How can you use this result in your own research or applications?

A: This result can be used in many different ways, depending on your research or applications. Some possible ways to use this result include:

• Studying the properties of closed linear operators and their domains
• Developing new results and theorems related to closed linear operators
• Applying this result to specific problems or applications in functional analysis

Conclusion

In conclusion, the result that $T$ is bounded if and only if $V$ is closed is an important result in functional analysis. It provides a necessary and sufficient condition for a closed linear operator to be bounded, and it has many applications in the study of closed linear operators and their properties. By understanding this result and its proof, you can gain a deeper understanding of the properties of closed linear operators and their domains.