Question

Let \(x, x_{n} \in \ell_{2}\) be such that \(x_{n} \stackrel{w}{\rightarrow} x\) in \(\ell_{2}\). Show that there is a subsequence \(\left\\{x_{n_{k}}\right\\}\) such that the Cesàro means $$ \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} $$ converge to \(x\) in \(\ell_{2}\) (the Banach-Saks theorem). Show that an analogous statement is true for \(c_{0}\). Spaces that have this property are called spaces with the weak BanachSaks property.

Short answer

From a weakly convergent sequence in \ \ell_{2} \ (or \ c_{0} \), one can extract a subsequence whose Cesàro means converge strongly to the limit.

Step-by-step solution

Understand the Weak Convergence

Given that a sequence \(\{x_{n}\}\) converges weakly to \(x\) in \(\ell_{2}\), i.e., \ x_{n} \stackrel{w}{\rightarrow} x \ in \(\ell_{2}\).

Define the Cesàro Mean

The Cesàro mean of a subsequence \(\{x_{n_{k}}\}\) is given by \ \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} \.

Construct a Subsequence Using the Banach-Saks Theorem

According to the Banach-Saks theorem, from the weakly convergent sequence \(\{x_{n}\}\), one can extract a subsequence \(\{x_{n_{k}}\}\) such that the Cesàro means \ \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} \ converge strongly to \(x\) in \(\ell_{2}\).

Verify Strong Convergence

To verify the Cesàro means of the subsequence \(\{x_{n_{k}}\}\), show that \ \lim_{k \to \infty} \left\| \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} - x \right\| = 0 \ in \(\ell_{2}\). Here we can use properties such as convexity and results involving weak convergence implying norm convergence in Cesàro means.

Analogous Statement for c_0

An analogous statement holds for the space \(c_{0}\). If \(x_{n} \stackrel{w}{\rightarrow} x\) in \(c_{0}\), then there exists a subsequence \(\{x_{n_{k}}\}\) such that the Cesàro means \ \frac{x_{n_{1}}+x_{n_{2}}+\ldots+x_{n_{k}}}{k} \ converge to \(x\) in \(c_{0}\). Spaces having this property are those with the weak Banach-Saks property.

Key concepts

Weak Convergence

In mathematics, specifically in functional analysis, weak convergence is a way in which a sequence of vectors can converge to a limit vector in a Hilbert or Banach space.
Unlike norm convergence where the sequence converges in norm, weak convergence is noted by the notation \(x_n \rightharpoonup x\).
For a sequence \(\{x_n\}\) in \(\ell_2\) space, weak convergence to a vector \(x\) means that for any \(y\) in the dual space, the inner product \(\langle x_n, y \rangle\) converges to \(\langle x, y \rangle\).
Weak convergence is weaker than norm convergence, meaning that if a sequence is norm convergent, it is also weakly convergent, but the converse is not necessarily true.
Weak convergence is crucial in analyzing and understanding the Cesàro mean convergence and properties of sequences in \(\ell_2\) and other spaces.

Cesàro Mean

The Cesàro mean is a method to analyze the convergence of sequences.
Instead of looking at the original sequence \(\{x_n\}\), the Cesàro mean considers the average of the first \(k\) terms.
For a sequence \(\{x_n\}\), the Cesàro mean of a subsequence \(\{x_{n_k}\}\) is given by \(\frac{x_{n_1} + x_{n_2} + \.\.\.+ x_{n_k}}{k}\).
This technique can smooth out fluctuations in the sequence, making it easier to demonstrate convergence. When applied to sequences in \(\ell_2\), the Cesàro mean often helps achieve results regarding norm convergence from weakly convergent sequences.

Weak Banach-Saks Property

The weak Banach-Saks property is a characteristic of certain Banach spaces.
It ensures the existence of a subsequence such that its Cesàro means converge in norm.
Specifically, for a weakly convergent sequence in a Banach space with this property, there is a subsequence where Cesàro means converge strongly to the limit.
This property is named after Stefan Banach and Stanisław Saks, who studied such types of convergence.
Knowing that a space has the weak Banach-Saks property provides a powerful tool in functional analysis, helping to bridge the gap between weak and norm convergence.

Norm Convergence

Norm convergence in functional analysis means that the sequence of vectors converges in norm to a limit vector.
For a sequence \(\{x_n\}\) in \(\ell_2\), norm convergence to a vector \(x\) is denoted by \( \lim_{n \to \infty} \| x_n - x \| = 0 \).
It is a stronger form of convergence compared to weak convergence.
Norm convergence plays an essential role in verifying the convergence of Cesàro means derived from weakly convergent sequences.
Establishing that Cesàro means of a subsequence converge in norm often involves demonstrating that the average of the sequence terms becomes arbitrarily close to the limit vector’s norm.

Subsequence Extraction

Subsequence extraction is a common technique in functional analysis used to analyze sequences.
When a sequence does not exhibit the desired convergence properties, extracting a subsequence can sometimes help.
By choosing appropriate terms, one may find a subsequence with better convergence properties.
In the context of the Banach-Saks theorem, subsequence extraction is used to find a subsequence whose Cesàro means converge in norm, even if the original sequence only converges weakly.
This technique is powerful and often employed in various areas of mathematical analysis to achieve stronger convergence results from weaker ones.