Question

Let \(X\) be a separable Banach space, and let \(\left\\{x_{i}\right\\}\) be dense in \(B_{X}\). Define an operator \(T\) from \(X^{*}\) into \(\ell_{2}\) by \(T(f)=\left(2^{-i} f\left(x_{i}\right)\right)\). Show that \(T\) is a homeomorphism of \(\left(B_{X^{*}}, w^{*}\right)\) onto the compact set \(T\left(B_{X^{*}}\right)\) in \(\ell_{2}\) taken in its norm topology.

Short answer

The operator \(T\) is a homeomorphism of \((B_{X^{*}}, w^{*})\) onto \(T(B_{X^{*}})\) in \( \ell_{2} \).

Step-by-step solution

- Define the Operator

Define the operator \(T\) from \(X^{*}\) into \(\ell_{2}\) by \(T(f)=\left(2^{-i} f\left(x_{i}\right)\right)\). Verify that this operator is well-defined.

- Check Operator Linear Property

Show that \(T\) is linear. By definition, \(T(af + bg) = (2^{-i} (af+bg)(x_i)) = a T(f) + b T(g)\) for any scalars \(a\) and \(b\).

- Ensure Compactness

Prove that \(T\) maps \(B_{X^{*}}\) to a compact subset of \(\ell_{2}\). Use the fact that \(B_{X^{*}}\) is \(weak^{*}\) compact and the mapping properties of \(T\).

- Verify Injectivity

Establish that \(T\) is injective. If \(T(f) = T(g)\), show that \(f = g\).

- Check Weak* Continuity

Verify that \(T\) is \(weak^{*}\) to norm continuous. That is, if \(f_{\alpha} \to f\) in the \(weak^{*}\) topology, \(T(f_{\alpha}) \to T(f)\) in the norm topology of \(\ell_{2}\).

- Establish Homeomorphism

Show that \(T\) maps \((B_{X^{*}}, w^{*})\) homeomorphically onto its image, using the results from Step 5 and the fact that \(B_{X^{*}}\) is \(weak^{*}\) compact.

Key concepts

Banach space

A Banach space is a complete normed vector space. This means any Cauchy sequence in this space converges within the space, which is important for stability in analysis. A common example is the space of continuous functions on a closed interval with the supremum norm. Banach spaces are fundamental in functional analysis because they provide a setting where limits and linear operations can be handled robustly.

Separable Banach spaces, like in the exercise, have a countable dense subset. This makes studying them simpler through sequences and ensuring various properties like compactness and continuity.

operator theory

Operator theory deals with bounded linear operators on functional spaces. These operators map elements from one space to another, maintaining the structure inherent to the spaces.

In our exercise, we analyze an operator denoted as \(T\) mapping from \(X^*\) (the dual space of \(X\)) into \(\ell_2\). The goal is to prove that this operator is a homeomorphism, which reflects both algebraic structure (linear behavior) and topological structure (continuity and compactness properties).

Key properties verified include linearity (i.e., \(T(af + bg) = a T(f) + b T(g)\)) and continuity, ensuring that the mapping respects the topological structure adequately.

compactness

Compactness in functional spaces generally means any sequence has a convergent subsequence, akin to closed and bounded sets in finite dimensions. In the exercise, we must show that \(T\) maps \(B_{X^*}\) to a compact subset of \(\ell_2\).

The unit ball \(B_{X^*}\) in the dual space, endowed with the weak* topology, is compact by the Banach-Alaoglu theorem. The operator's role is to preserve this compactness property under the transformation from \(B_{X^*}\) within \(X^*\) to its image in \(\ell_2\) while maintaining norm topology. This shows how deep connections between topology and linear operations are crucial in advanced functional analysis.

linear mappings

Linear mappings are functions preserving vector space operations, meaning \(T(a x + b y) = a T(x) + b T(y)\) for scalars \(a, b\) and vectors \(x, y\). This property is critical in operator theory since it ensures structural consistency when transforming spaces.

In our step-by-step solution, it was shown how \(T\) maintains linearity, a foundational property. Verifying this involves confirming that the output of \(T\) adheres to the same linear combinations as the input, thus preserving the linear structure exactly.

This linearity is not just a formality; it ensures that any analysis or manipulation conducted within the Banach space translates directly into similar operations within the image space.

weak* topology

The weak* topology is a specific topology on the dual space \(X^*\) of a Banach space \(X\). Here, a sequence \(f_\alpha\) converges to \(f\) if \(f_\alpha(x) \to f(x)\) for every \(x \in X\). This makes it coarser (fewer open sets) compared to the norm topology, meaning convergence in weak* is more relaxed.

In reviewing step 5, verifying weak* to norm continuity was crucial. It showed that if \(f_\alpha\) converges to \(f\) in weak* topology, \(T(f_\alpha)\) must converge to \(T(f)\) in the norm topology of \(\ell_2\).

Understanding the weak* topology and its relation to compactness (Banach-Alaoglu theorem) and continuity (operator \(T\)'s mapping properties) allows deeper insight into the interplay of topologies within functional spaces.