Question

Let \(X\) and \(Y\) be Banach spaces. Show that if a linear operator \(T\) from \(X\) into \(Y\) is \(w-w\) -continuous, then \(T \in \mathcal{B}(X, Y) .\) On the other hand, every bounded linear operator is \(w-w\) -continuous.

Short answer

If \( T \) is \( w-w \)-continuous, then \( T \) is bounded. Every bounded linear operator is \( w-w \)-continuous.

Step-by-step solution

Define Terms

Start by defining the terms. A linear operator from one Banach space to another is denoted as \( T: X \to Y \). A linear operator \( T \) is \( w-w \)-continuous if it is continuous from the weak topology on \( X \) to the weak topology on \( Y \). \( T \in \mathcal{B}(X, Y) \) means \( T \) is a bounded linear operator.

Show \( T \) is Bounded

To show that if \( T \) is \( w-w \)-continuous, then \( T \in \mathcal{B}(X, Y) \), assume \( T \) is \( w-w \)-continuous. Let \( \{x_n\} \subset X \) be such that \( x_n \to 0 \) weakly. Since \( T \) is \( w-w \)-continuous, it maps weakly convergent sequences to weakly convergent sequences. Specifically, \( T(x_n) \to T(0)=0 \) weakly in \( Y \).

Use Banach-Steinhaus Theorem

Apply the Banach-Steinhaus theorem (uniform boundedness principle): If a family of bounded linear operators \( \{T_n\} \) from \( X \) to \( Y \) is pointwise bounded (for every \( x \in X \), the sequence \( \{ T_n(x) \} \) is bounded), then the family is uniformly bounded. Given the weak compactness and weak-to-weak continuity, use this theorem to conclude that \( T \) is bounded, thus \( T \in \mathcal{B}(X, Y) \).

Prove the Reverse

To show the reverse, that every bounded linear operator is \( w-w \)-continuous, let \( T \in \mathcal{B}(X, Y) \). For any weakly convergent sequence \( x_n \to x \) in \( X \), show that \( T(x_n) \to T(x) \) weakly in \( Y \) by using the fact that bounded linear operators are continuous under the weak topologies of Banach spaces.

Key concepts

Banach space

A Banach space is a complete normed vector space. This means every Cauchy sequence (a sequence where elements get arbitrarily close to each other) in the space converges to an element within the space. Understanding Banach spaces is crucial in functional analysis because they provide a well-defined context for analyzing continuous linear operators.
Banach spaces are named after the Polish mathematician Stefan Banach.
In mathematics, they help describe spaces where we can extend calculus beyond finite dimensions.

weak topology

In functional analysis, the weak topology on a Banach space is a type of topology that is coarser (less fine) than the norm topology. In other words, it has fewer open sets.
A sequence in a Banach space is said to converge weakly if it converges in the weak topology. This means that instead of requiring that the sequence converges in norm (which is stronger), we require that it converges when tested against all continuous linear functionals.
One important aspect is that while weak convergence is less stringent than norm convergence, it still preserves essential properties of sequences and functions in the space.

Banach-Steinhaus theorem

The Banach-Steinhaus theorem is also known as the uniform boundedness principle. It is a fundamental result in functional analysis which states that for a family of bounded linear operators from a Banach space into a normed vector space, pointwise boundedness implies uniform boundedness.
What this means is: if a family of operators does not make any single element in the space infinitely large when applied, then there is a common bound across the entire family of operators. This theorem is critical for proving properties about bounded linear operators and ensuring that our manipulations on these operators retain controlled behavior.

weak continuity

A linear operator is said to be weakly continuous if it preserves weak convergence. In practical terms, this means if a sequence converges weakly to some point in a Banach space, applying this operator to the sequence will produce a new sequence that also converges weakly to an image point in another Banach space.
Weak continuity is intimately connected with the weak topology. If a linear operator is weakly continuous, it implies certain bounded behaviors due to weaker forms of convergence. This is why proving weak continuity often involves using tools like the Banach-Steinhaus theorem to connect weak behaviors to norm-bounded properties.