Question

Let \(\left\\{x_{n}\right\\}\) be a sequence in a Banach space \(X\). Prove that \(x_{n} \stackrel{w}{\rightarrow} x\) if and only if \(\left\\{x_{n}\right\\}\) is bounded and the set \(\left\\{f \in X^{*} ; f\left(x_{n}\right) \rightarrow f(x)\right\\}\) is dense in \(X^{*}\). Similarly, let \(\left\\{f_{n}\right\\}\) be a sequence in \(X^{*}\). Prove that \(f_{n} \stackrel{w^{*}}{\rightarrow} f\) if and only if \(\left\\{f_{n}\right\\}\) is bounded and the set \(\left\\{x \in X ; f_{n}(x) \rightarrow f(x)\right\\}\) is dense in \(X\). In particular, we have the following corollary. Assume that \(\left\\{x_{n}\right\\}\) is bounded and \(M \subset X^{*}\) is such that \(\overline{\operatorname{span}}(M)=X^{*}\) and \(f\left(x_{n}\right) \rightarrow f(x)\) for all \(f \in M .\) Then \(x_{n} \stackrel{w}{\rightarrow} x .\) An analogous statement is true for \(w^{*}\) -convergence.

Short answer

Weak convergence holds if the sequence is bounded and functionals are dense. The same applies to weak star convergence.

Step-by-step solution

Understanding Weak Convergence

Weak convergence, denoted by \(x_{n} \stackrel{w}{\rightarrow} x\), means that for every functional f in the dual space X*, \ f(x_{n}) \rightarrow f(x)\.

Show Necessary Conditions for Weak Convergence

Assume \(x_{n} \stackrel{w}{\rightarrow} x\). First, show that \$\{x_{n}\}$$\ is bounded. By definition of weak convergence, \$|f(x_{n})| \leq C\ for all f \in X^*$$\ and some constant C.

Density Condition

Weak convergence implies for a dense set of functionals \(f \in X^{*}\), \ f(x_{n}) \rightarrow f(x)\.

Sufficient Conditions for Weak Convergence

Now, assume \$\{x_{n}\}$$\ is bounded and the set \$\{f \in X^* ; f(x_{n}) \rightarrow f(x)\}\$$\ is dense in \$X^*$$. This implies \$f(x_{n}) \rightarrow f(x)\ for all f \in X^*$$\ because every element of \$X^*$$ can be approximated by elements of the dense set.

Weak Star Convergence

\f_{n} \stackrel{w^{*}}{\rightarrow} f\ if and only if \$\{f_{n}\}$$\ is bounded and the set \$\{x \in X ; f_{n}(x) \rightarrow f(x)\}\$$\ is dense in X. Use the analogous steps as for weak convergence, noting that weak star convergence pertains to the dual space X*.

Corollary for Weak Convergence

Given \$\{x_{n}\}$$\ is bounded and \$M \subset X^{*}\$$ with \$\overline{\operatorname{span}}(M)=X^{*}\$$, and \$f(x_{n}) \rightarrow f(x)\ for all f \in M\$$. This implies \$x_{n} \stackrel{w}{\rightarrow} x\$$ due to the density of M and bounded sequence.

Corollary for Weak Star Convergence

By analogy, if \$\{f_{n}\}$$\ is bounded and \$N \subset X\$$ with \$\overline{\operatorname{span}}(N)=X\$$ and \$f_{n}(x) \rightarrow f(x)\$$ for all x \in N\, then \$f_{n} \stackrel{w^{*}}{\rightarrow} f\$$ also follows.

Key concepts

Banach space

A Banach space is a special type of vector space. It is both complete and normed.

*Complete* means that any Cauchy sequence of points in this space converges to a point within the space. This property is crucial in analysis because it ensures that limits of sequences that behave well (in terms of their distance) will also be within our space.

**Normed** means that each vector in the space has a length, known as a norm, which is a non-negative real number. This norm behaves in a way we generally expect length to behave:

• It is zero if and only if the vector is the zero vector.
• It scales linearly with the vector.
• It satisfies the triangle inequality, meaning the norm of the sum of two vectors is less than or equal to the sum of their norms.

These two properties together make Banach spaces particularly useful because they provide both a measure of 'size' and the guarantee that sequences with certain properties will have limits within the space.

Dual space

The dual space of a Banach space X, denoted X*, is another Banach space consisting of all continuous linear functionals on X.

A *linear functional* is a special type of function that takes a vector from X and assigns it a scalar (a real or complex number) in such a way that it respects the linear structure of X. This means it satisfies:

• Linearity: f(ax + by) = af(x) + bf(y) for any scalars a and b, and vectors x and y in X.
• Continuity: small changes in the vector result in small changes in the functional's value.

The dual space inherits a norm from X and is complete under this norm, making it also a Banach space. This structure is important because dual spaces allow us to study the original Banach space using tools from analysis.

Weak star convergence

Weak star convergence is a mode of convergence for functionals in the dual space X*.

We say a sequence of functionals \(f_n\) in X* weak star converges to a functional f in X* (written as \(f_n \rightarrow f \text{ weak}^*\)) if, for every vector x in X, we have: \lim_{n \to \infty} f_n(x) = f(x)\.

This notion of convergence is weaker than norm convergence, where f_n actually becomes close to f in norm. Weak star convergence only requires that the sequence of values \(f_n(x)\) approximates \(f(x)\) for each vector x in X.

This type of convergence is important because it often allows for more sequences to converge compared to norm convergence. It is particularly useful in the context of bounded sequences of functionals, as it lays a framework to discuss convergence without reference to the norm of the functionals themselves.

Density

In the context of Banach spaces and their duals, density refers to the property of a subset M of a space X such that every element of X can be approximated arbitrarily well by elements of M.

More formally, a set M is dense in X if for every element x in X and every epsilon > 0, there is an element m in M such that the distance between x and m is less than epsilon.

For example, the rationals are dense in the real numbers because any real number can be approximated by rationals to any degree of accuracy.

In our context, density helps us understand convergence. If functionals defined on a dense set converge to zero on that set, they tend to zero everywhere in the space, because the dense subset approximates the entire space.

Bounded sequence

A sequence \(x_n\) in a Banach space X is bounded if there exists a constant C such that for all n, the norm \|x_n\ \( \leq C\).

This means that all terms of the sequence remain within a 'ball' of radius C centered at the origin of the space.

Boundedness is significant in analysis because it often ensures that sequences have subsequences that converge (via the Banach-Alaoglu theorem in the weak* topology or Bolzano-Weierstrass theorem in normed spaces). For example, the weak convergence defined in the context requires the sequences to be bounded. Without this requirement, weak convergence may not hold.