Equivalent Formulations For Nets And Weak Convergence In A Normed Vector Space.

Introduction

In the realm of functional analysis, the study of normed vector spaces and their properties is a fundamental area of research. One of the key concepts in this field is the notion of weak convergence, which is a crucial tool in understanding the behavior of sequences and nets in normed spaces. In this article, we will explore the equivalent formulations for nets and weak convergence in a normed vector space, with a focus on the set of all bounded linear maps that factor through a finite-dimensional space.

Preliminaries

Let $X$ be a normed real vector space, and let $\mathscr{A}$ be the set of all bounded linear maps $T$ that factor as { X\xrightarrow{\text{T_1}} (\mathbb{R}^n,\lVert \cdot \rVert_1)\xrightarrow{\text{T_2}} ... \xrightarrow{\text{T_k}} (\mathbb{R}^m,\lVert \cdot \rVert_m) } where $n,m \in \mathbb{N}$, and $T_i$ are bounded linear maps for each $i$. We will denote this set as $\mathscr{A}_n$ for each $n \in \mathbb{N}$.

Nets and Weak Convergence

A net in a normed space $X$ is a function $f: \Lambda \to X$ from a directed set $\Lambda$ to $X$. A net $f$ is said to converge weakly to an element $x \in X$ if for every bounded linear functional $T \in X^*$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ We will denote this as $f \xrightarrow{w} x$.

Equivalent Formulations

We will now present the equivalent formulations for nets and weak convergence in a normed vector space.

Theorem 1: Nets and Weak Convergence

Let $X$ be a normed real vector space, and let $f: \Lambda \to X$ be a net. Then the following are equivalent:

• $f$ converges weakly to an element $x \in X$, i.e., $f \xrightarrow{w} x$.
• For every $T \in \mathscr{A}$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$

Proof

• $(i) \Rightarrow (ii)$: Let $f \xrightarrow{w} x$. Then for every $T \in X^*$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ Since $\mathscr{A} \subset X^*$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ for every $T \in \mathscr{A}$.
• $(ii) \Rightarrow (i)$: Let $T \in \mathscr{A}$ be arbitrary. Then we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ for every $T \in \mathscr{A}$. Since $\mathscr{A}$ is dense in $X^*$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ for every $T \in X^*$. Therefore, $f \xrightarrow{w} x$.

Theorem 2: Weak Convergence and Nets

Let $X$ be a normed real vector space, and let $f: \Lambda \to X$ be a net. Then the following are equivalent:

• $f$ converges weakly to an element $x \in X$, i.e., $f \xrightarrow{w} x$.
• For every $n \in \mathbb{N}$, we have ${ \lim_{\lambda \in \Lambda} T_n(f(\lambda)) = T_n(x) }$ where $T_n \in \mathscr{A}_n$ for each $n$.

Proof

• $(i) \Rightarrow (ii)$: Let $f \xrightarrow{w} x$. Then for every $T \in X^*$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ Since $\mathscr{A}_n \subset X^*$ for each $n$, we have ${ \lim_{\lambda \in \Lambda} T_n(f(\lambda)) = T_n(x) }$ for every $T_n \in \mathscr{A}_n$.
• $(ii) \Rightarrow (i)$: Let $T_n \in \mathscr{A}_n$ be arbitrary for each $n$. Then we have ${ \lim_{\lambda \in \Lambda} T_n(f(\lambda)) = T_n(x) }$ for every $n \in \mathbb{N}$. Since $\mathscr{A}_n$ is dense in $X^*$ for each $n$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ for every $T \in X^*$. Therefore, $f \xrightarrow{w} x$.

Conclusion

In this article, we have presented the equivalent formulations for nets and weak convergence in a normed vector space. We have shown that a net $f$ converges weakly to an element $x \in X$ if and only if for every $T \in \mathscr{A}$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ We have also shown that a net $f$ converges weakly to an element $x \in X$ if and only if for every $n \in \mathbb{N}$, we have ${ \lim_{\lambda \in \Lambda} T_n(f(\lambda)) = T_n(x) }$ where $T_n \in \mathscr{A}_n$ for each $n$. These results provide a deeper understanding of the relationship between nets and weak convergence in normed vector spaces.

References

• [1] Kelley, J. L. (1955). General Topology. Van Nostrand.
• [2] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
• [3] Yosida, K. (1980). Functional Analysis. Springer-Verlag.

Further Reading

For further reading on the topic of nets and weak convergence in normed vector spaces, we recommend the following resources:

• [1] Albiac, F., & Kalton, N. J. (2006). Topics in Banach Space Theory. Springer-Verlag.
• [2] Diestel, J. (1975). Sequences and Series in Banach Spaces. Springer-Verlag.
• [3] Lindenstrauss, J., & Tzafriri, L. (1977). Classical Banach Spaces I. Springer-Verlag.

Note: The references provided are a selection of the most relevant and influential works in the field of functional analysis and normed vector spaces.

Introduction

In our previous article, we explored the equivalent formulations for nets and weak convergence in a normed vector space. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the difference between a net and a sequence in a normed vector space?

A: A net in a normed vector space is a function from a directed set to the space, while a sequence is a special type of net where the directed set is the natural numbers with the usual ordering.

Q: Why is weak convergence important in functional analysis?

A: Weak convergence is important in functional analysis because it provides a way to study the behavior of sequences and nets in normed spaces without requiring the use of the norm. This is particularly useful in situations where the norm is not well-defined or is difficult to work with.

Q: What is the relationship between nets and weak convergence in a normed vector space?

A: As we showed in our previous article, a net $f$ converges weakly to an element $x \in X$ if and only if for every $T \in \mathscr{A}$, we have ${ \lim_{\lambda \in \Lambda} T(f(\lambda)) = T(x) }$ This means that weak convergence is equivalent to the convergence of the net under the action of all bounded linear functionals.

Q: Can you give an example of a net that converges weakly but not strongly in a normed vector space?

A: Yes, consider the sequence of functions $f_n(x) = x^n$ in the space $C[0,1]$ of continuous functions on the interval $[0,1]$. This sequence converges weakly to the zero function, but it does not converge strongly in the norm of the space.

Q: How do you determine whether a net converges weakly or strongly in a normed vector space?

A: To determine whether a net converges weakly or strongly, you need to check whether the net converges under the action of all bounded linear functionals or whether it converges in the norm of the space.

Q: Can you give an example of a normed vector space where weak convergence is not equivalent to strong convergence?

A: Yes, consider the space $l^1$ of absolutely convergent sequences. In this space, weak convergence is not equivalent to strong convergence.

Q: What is the significance of the set $\mathscr{A}$ of all bounded linear maps that factor through a finite-dimensional space?

A: The set $\mathscr{A}$ is significant because it provides a way to study the behavior of nets and weak convergence in normed spaces without requiring the use of the norm. The set $\mathscr{A}$ is dense in the space of all bounded linear functionals, which means that any bounded linear functional can be approximated by a map in $\mathscr{A}$.

Q: Can you give an example of a normed vector space where the set $\mathscr{A}$ is not dense in the space of all bounded linear functionals?

A: Yes, consider the space $C[0,1]$ of continuous functions on the interval $[0,1]$. In this space, the set $\mathscr{A}$ is not dense in the space of all bounded linear functionals.

Q: What is the relationship between the set $\mathscr{A}$ and the space of all bounded linear functionals?

A: The set $\mathscr{A}$ is dense in the space of all bounded linear functionals, which means that any bounded linear functional can be approximated by a map in $\mathscr{A}$.

Q: Can you give an example of a normed vector space where the set $\mathscr{A}$ is not dense in the space of all bounded linear functionals?

A: Yes, consider the space $l^1$ of absolutely convergent sequences. In this space, the set $\mathscr{A}$ is not dense in the space of all bounded linear functionals.

Conclusion

In this article, we have answered some of the most frequently asked questions about the equivalent formulations for nets and weak convergence in a normed vector space. We hope that this article has provided a helpful resource for those interested in this topic.

References

• [1] Kelley, J. L. (1955). General Topology. Van Nostrand.
• [2] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
• [3] Yosida, K. (1980). Functional Analysis. Springer-Verlag.

Further Reading

For further reading on the topic of nets and weak convergence in normed vector spaces, we recommend the following resources:

• [1] Albiac, F., & Kalton, N. J. (2006). Topics in Banach Space Theory. Springer-Verlag.
• [2] Diestel, J. (1975). Sequences and Series in Banach Spaces. Springer-Verlag.
• [3] Lindenstrauss, J., & Tzafriri, L. (1977). Classical Banach Spaces I. Springer-Verlag.