Question

Let \(X\) be a separable Banach space, and let \(\left\\{x_{n}\right\\}\) be a dense sequence in \(S_{X}\). Define a mapping \(T\) from \(X^{*}\) into \(\ell_{2}\) by \(T(f)=\left(f\left(x_{i}\right) / 2^{i}\right)\). Show that \(T\) is a bounded linear operator that is continuous from the \(w^{*}\) -topology of \(X^{*}\) into the \(w\) -topology of \(\ell_{2}\).

Short answer

T is a bounded linear operator, continuous from the w*-topology of X* to the w-topology of ℓ2.

Step-by-step solution

- Define the mapping and linearity

The mapping is given by: \[ T(f) = \left( \frac{f(x_i)}{2^i} \right). \] Show that this mapping is linear by verifying that for any \(f, g \in X^*\) and scalars \(a, b\), we have: \[ T(af+bg) = \left( \frac{(af+bg)(x_i)}{2^i} \right) = a \left( \frac{f(x_i)}{2^i} \right) + b \left( \frac{g(x_i)}{2^i} \right) = aT(f) + bT(g). \]

- Show boundedness

To show \(T\) is bounded, find a constant \(C\) such that \( \| T(f) \|_{\ell_2} \leq C \| f \|_{X^*} \). First, evaluate the \(\ell_2\)-norm of \(T(f)\): \[ \| T(f) \|_{\ell_2} = \sqrt{ \sum_{i=1}^{\infty} \left( \frac{f(x_i)}{2^i} \right)^2 }. \] Use the fact that \( \| f(x_i) \| \leq \| f \|_{X^*} \). Then, \[ \| T(f) \|_{\ell_2} \leq \| f \|_{X^*} \sqrt{ \sum_{i=1}^{\infty} \left( \frac{1}{2^i} \right)^2 }. \] Calculate the sum, \[ \sum_{i=1}^{\infty} \left( \frac{1}{2^i} \right)^2 = \sum_{i=1}^{\infty} \frac{1}{4^i} = \frac{1/4}{1-1/4} = \frac{1/4}{3/4} = \frac{1}{3}. \] So, \[ \| T(f) \|_{\ell_2} \leq \| f \|_{X^*} \sqrt{ \frac{1}{3} } = \frac{ \| f \|_{X^*} }{ \sqrt{3} }. \] Therefore, \(T\) is bounded with \(C = \frac{1}{\sqrt{3}}\).

- Continuity with w*-topology to w-topology

Show that \(T\) is continuous from the \(w^{*}\)-topology of \(X^{*}\) to the \(w\)-topology of \(\ell_2\). Let \(f_\alpha\) be a net in \(X^*\) that converges to \(f\) in the \(w^{*}\) topology. This means for all \(x \in X\), \(f_\alpha(x)\) converges to \(f(x)\). To show the \(w\)-continuity, we want to show that \(\left( T(f_\alpha) \right) \) converges to \(T(f)\) in the weak topology of \(\ell_2\). For any \(y = (y_i) \in \ell_2\), consider: \[ \left< T(f_\alpha), y \right> = \sum_{i=1}^{\infty} \frac{f_\alpha(x_i)}{2^i} y_i. \] By the \(w^{*}\)-convergence, \( f_\alpha(x_i) \to f(x_i) \) for each \(i\). Therefore, \[ \left< T(f_\alpha), y \right> \to \sum_{i=1}^{\infty} \frac{f(x_i)}{2^i} y_i = \left< T(f), y \right>. \] Hence, \( T(f_\alpha) \to T(f) \) in the \(w\)-topology of \(\ell_2\).

Key concepts

Banach space

A Banach space is a complete normed vector space. This means that every Cauchy sequence in the space converges within the space.
Banach spaces are fundamental in functional analysis and have applications in various areas such as differential equations, quantum mechanics, and signal processing.

A typical example of a Banach space is the space of continuous functions on a closed interval, with the maximum norm. When reading about Banach spaces, you might often come across these key points:

• Completeness: Every Cauchy sequence has a limit within the space.
• Normed: There is a defined way to measure the length (or norm) of vectors in the space.
• Linear: The space includes operations of vector addition and scalar multiplication.

Weak topology

In functional analysis, the weak topology on a normed space is the coarsest topology in which all continuous linear functionals remain continuous.
This means that a sequence (or more generally, a net) converges in the weak topology if and only if it converges pointwise with respect to all continuous linear functionals.

An intuitive way to understand weak topology is to think of it as a 'weaker' form of the usual topology (norm topology) because it has fewer open sets. Here are some essential aspects of weak topology:

• Pointwise Convergence: A sequence converges weakly if it converges in value for every linear functional.
• Fewer Open Sets: Leads to possibly larger sets of weakly convergent sequences.
• Applications: Used mostly in spaces of functions and discussions around dual spaces.

Weak-* topology

The weak-* topology is a topology on the dual space of a normed vector space. It is the weakest (coarsest) topology for which the evaluation maps are continuous.
If a sequence of functionals converges in the weak-* topology, then it converges pointwise on the original normed space. Understanding the weak-* topology often involves these ideas:

• Evaluation Continuity: The topology is built so that evaluations at points are continuous.
• Pointwise Convergence: Similar to weak topology, the functional sequences converge on evaluation.
• Banach-Alaoglu Theorem: A vital result that states the unit ball in the dual space is compact in the weak-* topology.

Continuous mapping

A continuous mapping between topological spaces preserves the notion of 'closeness'. In simpler terms, it ensures that small changes in the input lead to small changes in the output.
More formally, a function between two topological spaces is continuous if the preimage of every open set in the target space is open in the source space.

When discussing continuous maps in the context of the weak and weak-* topology, some crucial points include:

• Topology Preservation: Maps that respect the defined topology structure.
• Convergence: The image of a converging sequence (or net) should also converge.
• Functional Analysis: Often involves spaces of functions and operators, leading to practical applications.