Question

Assume that the codimension of \(X\) in \(X^{* *}\) is finite. Show that then every \(w^{*}\) -dense closed subspace in \(X^{*}\) is norming.

Short answer

Any weak* dense closed subspace in X* is norming because X is densely embedded in X** and Y approximates X**.

Step-by-step solution

Understand the Problem Statement

The codimension of a subspace refers to the dimension of the quotient space formed by the original space and the subspace. Given that the codimension of the space X in its bidual space X** is finite, it's required to show that any closed subspace in the dual space X* which is dense in the weak* topology is also norming.

Define Key Terms

Identify the definitions: A subspace is norming if for every element x in X, the norm of x can be determined by supremum of the functional values over the norming subspace. A subspace is dense in the weak* topology if every element of X* can be approximated by elements of this subspace.

Use Duality and Weak* Density

Let Y be a weak* dense closed subspace in X*. We need to show that for every x in X, \(\|x\| = \sup_{\phi \in Y, \|\phi\| \leq 1} |\phi(x)|\). Since Y is weak* dense, for any element in X**, the values can be approximated by elements in Y.

Apply Codimension Finite Assumption

Since the codimension of X in X** is finite, this implies that X is dense in X** with the topology inherited from X**. Thus, any element of X** can be closely approximated by elements of X, making X** effectively norming.

Conclusion

Since Y is weak* dense in X* and X is dense in X**, it follows that Y must be norming for X. Therefore, every weak* dense closed subspace in X* is norming.

Key concepts

Codimension

Codimension is a fundamental idea in functional analysis. It's the difference in dimension between a larger space and a subspace. Basically, if you have a space, say \(X\), and a subspace within it, the codimension tells you how much 'smaller' the subspace is. Mathematically, if \(W\) is a subspace of a vector space \(V\), the codimension of \(W\) in \(V\) is \( \text{dim}(V/W) \). This entails calculating how many independent directions in \(V\) are not captured by \(W\).
In simpler terms, imagine having two rooms, one larger than the other. The larger room is \(X^{* *}\), and the smaller one is \(X\). The codimension is essentially the space not covered by \(X\).
In the context of the exercise, we are given that the codimension is finite, indicating a specific, manageable difference between the dimensions of \(X\) and \(X^{* *}\). This finiteness is key to proving the subsequent properties linked to weak* dense subspaces.

Weak* Topology

The weak* topology is an extension of the weak topology but focused on dual spaces. For a dual space \(X^{*}\), the weak* topology is the topology of pointwise convergence on \(X\). This means that a sequence of functionals \(\{\phi_n\}\) in \(X^{*}\) converges to a functional \(\phi\) in the weak* topology if and only if, for every \(x \in X\), \(\phi_n(x)\) converges to \(\phi(x)\).
An important property of this topology is that it is coarser than the norm topology, meaning fewer sets are open. This often simplifies certain proofs and analysis.
In our exercise, understanding weak* density is critical. A subspace \(Y\) is said to be weak*-dense in \(X^{*}\) if every functional in \(X^{*}\) can be approximated arbitrarily well by functionals from \(Y\). This approximation is crucial for demonstrating that weak*-dense closed subspaces are norming, as the weak* topology allows us to link the behavior of elements in \(X^{*}\) closely with elements in \(X\).

Norming Subspace

A norming subspace is a subset of a dual space that can 'measure' the norm of elements in the original space. If \(Y\) is a subspace of \(X^{*}\), it's norming if for every \(x \in X\), you can express the norm of \(x\) as \[ \|x\| = \sup_{\phi \in Y, \|\phi\| \leq 1} |\phi(x)| \].
This concept helps to 'pin' down the size of the original elements using the functionals in the norming subspace. If all elements in \(X\) can be evaluated by \(Y\), it implies \(Y\) contains enough information to fully describe the behavior of \(X\).
In the problem, proving that every weak*-dense closed subspace in \(X^{*}\) is norming involves showing that you can gauge the norm of any element in \(X\) using functionals from any such subspace. This is linked back to the finite codimension and the density properties in weak* topology, ensuring that these subspaces have enough coverage to serve as norming tools.

Duality

Duality is a powerful concept in functional analysis that connects a vector space with its dual space. For a given space \(X\), its dual, denoted \(X^{*}\), consists of all continuous linear functionals on \(X\). This relationship allows one to study properties of \(X\) through the lens of \(X^{*}\).
The notion extends to bidual spaces (\(X^{**}\)), encompassing functionals on the dual space \(X^{*}\). Often, \(X\) can be embedded in \(X^{**}\) via the canonical map \(J: X \to X^{**}\) given by \(J(x)(\phi) = \phi(x)\) for \(x \in X\) and \(\phi \in X^{*}\).
Our exercise leverages duality by showing how properties of finite codimension and weak* dens**ity** interplay to ensure norming characteristics. Since dual spaces intrinsically carry the structure of the original space, this deep connectivity helps prove that weak*-dense closed subspaces indeed norm. Remember, duality isn't just a theoretical construct—it's a practical tool that simplifies complex problems by switching perspectives.