Question

Let \(X\) be a separable Banach space. Find \(T \in \mathcal{B}\left(\ell_{2}, X\right)\) such that \(T\left(\ell_{2}\right)\) is dense in \(X .\) Note that then \(T^{*}: X^{*} \rightarrow \ell_{2}\) is a bounded linear \(w^{*}-w\) -continuous one-to-one operator.

Short answer

Construct \(T((a_n)) = \sum_{n=1}^{\infty} a_n x_n\), ensuring the image is dense. The adjoint \(T^*\) is one-to-one and weak*-weak continuous.

Step-by-step solution

- Understand the Problem

We need to find a bounded linear operator from \(\ell_2\) to a separable Banach space \(X\) such that the image of \(\ell_2\) under this operator is dense in \(X\). Additionally, we need to analyze the adjoint operator \(T^*\) regarding its properties.

- Review Dense Sets

Recall that a subset \(A\) of a space \(X\) is dense if every point in \(X\) is a limit point of \(A\) or belongs to \(A.\) For \(T(\ell_2)\) to be dense in \(X\), every point in \(X\) must be approximable by points in the image \(T(\ell_2).\)

- Constructing the Operator \(T\)

Since \(X\) is separable, it has a dense countable subset. Let \(\{x_n\}_{n=1}^{\infty}\) be a dense sequence in \(X\). Define \(T: \ell_2 \rightarrow X\) by \(T((a_n)) = \sum_{n=1}^{\infty} a_n x_n,\) ensuring the map is well-defined and bounded.

- Checking Density

To verify \(T(\ell_2)\) is dense, note that any point of \(X\) can be approximated by linear combinations of \(\{x_n\}.\) Given any \(x \in X\), find a sequence in \(T(\ell_2)\) converging to \(x\) since \(\{x_n\}\) is dense.

- Adjoint Operator \(T^*\)

For the adjoint operator \(T^* : X^* \rightarrow \ell_2\), defined by \(T^*(f) = (f(x_n))_{n=1}^{\infty}\), ensure it is linear and bounded. Check one-to-one property: \(T^*(f) = 0\) implies \(f(x_n) = 0\) for all \(n\). Since \(\{x_n\}\) is dense, \(f = 0\). Verify that \(T^*\) is also weak*-weak continuous.

Key concepts

Separable Banach Space

A Banach space is a complete normed vector space. The term 'separable' means that this space contains a countable, dense subset.
For instance, if we have a separable Banach space, we can find a sequence of elements within it that are so close to every element of the space that you can approximate any element as closely as desired using this sequence.

Think of it as trying to describe any number on the number line using only fractions with integers in their numerators and denominators. Even though there are infinitely many fractions, they can 'cover' the entire number line. This makes computations and proofs easier.

Dense Sets

A set is called dense in space if every point in the space is either in the set itself or arbitrarily close to some point in the set.
Mathematically speaking, a subset \(A\) of a space \(X\) is dense if for every point \(x \in X\), either \(x\) is in \(A\), or \(x\) can be approximated by points in \(A\).

In simpler terms, if you can pick any point in space and always find members of a dense set that get as close as you like to this point, the set is dense. This concept is vital in analysis because it helps us work with infinite spaces using more manageable countable subsets.

Adjoint Operator

In functional analysis, operators have 'adjoint' counterparts that essentially act in an 'inverse' manner in a dual space.
If you have an operator \(T\) from one Banach space to another, its adjoint \(T^*\) goes in the reverse direction but between their dual spaces \( X^* \) and \(Y^* \).

The adjoint operator has properties similar to those of the original operator but in the context of the dual spaces. It is particularly useful for ensuring certain kinds of continuity, such as weak-star continuity.

Weak-Star Continuity

This type of continuity is specialized for dual (topological) spaces where it considers the weak-* topology.
An operator is weak-star continuous if the convergence in the weak-* topology in one space implies convergence in the weak-* topology in another space. This is a bit tricky but can be simplified.

If you think of weak-* continuity, it means that elements' effects, through functional evaluations, behave nicely with sequences. For instance, if a functional 'almost' converges, weak-star continuity assures you can take this 'almost' convergence through your operator without issues.

Linear Combinations

The concept of linear combinations forms the backbone of vector spaces.
A linear combination of a set of elements is an expression created by multiplying each element by a corresponding scalar and summing the results. For example, if you have vectors \(v_1, v_2, ..., v_n\) and scalars \(a_1, a_2, ..., a_n\), the term \(a_1v_1 + a_2v_2 + ... + a_nv_n\) is a linear combination.

This concept is crucial as it allows the construction of new vectors (or points) within the space, and in our context, it helps in demonstrating how dense sets can approximate any element in space efficiently.