Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\mathcal{X}''$ of the Banach space $(\mathcal{X}',\|\cdot\|_{\mathcal{X}'})$. The unit ball of $\mathcal{X}'$ is denoted by $$\mathcal{B} = \{ y \in \mathcal{X}', \ \| y\|_{\mathcal{X}'} \leq 1\}.$$

We consider three topologies on $\mathcal{X}'$, on which we recap basic facts:

• The norm topology, for which $\mathcal{B}$ is not compact as soon as $\mathcal{X}$ is infinite dimensional (Riesz' theorem).
• The weak* topology, which is the coarsest topology such that the linear functionals $y \mapsto y(x)$ are continuous for any $x \in \mathcal{X}$. The Banach-Alaoglu theorem states that $\mathcal{B}$ is compact for the weak*-topology.
• The weak topology, which is the coarsest topology such that the linear functionals $y \mapsto z(y)$ are continuous for $z \in \mathcal{X}''$.

The weak* topology is weaker than the weak topology, which is weaker than the norm topology. Moreover, the unit ball $\mathcal{B}$ is not compact for the weak topology as soon as the space is not reflexive (otherwise, the weak and weak* topologies coincide).

My question is the following: Are there intermediate topologies between the weak* and the weak topology for which the unit ball $\mathcal{B}$ is compact? Or can we say in some sense that the weak* topology is the finest for which the unit ball is compact?

If it helps, the same questions can be considered for the specific cases:

• $(\mathcal{X},\mathcal{X}',\mathcal{X}'') = (c_0(\mathbb{Z}), \ell_1(\mathbb{Z}), \ell_\infty(\mathbb{Z}))$ where $c_0(\mathbb{Z})$ is the space of vanishing sequences endowed with the norm $\|\cdot\|_\infty$.
• $(\mathcal{X},\mathcal{X}',\mathcal{X}'') = (\mathcal{C}(\mathbb{T}), \mathcal{M}(\mathbb{T}), \mathcal{M}'(\mathbb{T}))$ where $\mathcal{T}$ is the torus, $\mathcal{C}(\mathbb{T})$ the space of continuous periodic functional endowed with the supremum norm, and $\mathcal{M}(\mathbb{T}) the space of finite Radon measure.

(Motivation: I try to understand what is the largest topology for which $\mathcal{B}$ is compact beyond the weak* topology in order to use the Krein-Millmann theorem ensuring the existence of extreme points for convex compact sets.)

Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\mathcal{X}''$ of the Banach space $(\mathcal{X}',\|\cdot\|_{\mathcal{X}'})$. The unit ball of $\mathcal{X}'$ is denoted by $$\mathcal{B} = \{ y \in \mathcal{X}', \ \| y\|_{\mathcal{X}'} \leq 1\}.$$

We consider three topologies on $\mathcal{X}'$, on which we recap basic facts:

• The norm topology, for which $\mathcal{B}$ is not compact as soon as $\mathcal{X}$ is infinite dimensional (Riesz' theorem).
• The weak* topology, which is the coarsest topology such that the linear functionals $y \mapsto y(x)$ are continuous for any $x \in \mathcal{X}$. The Banach-Alaoglu theorem states that $\mathcal{B}$ is compact for the weak*-topology.
• The weak topology, which is the coarsest topology such that the linear functionals $y \mapsto z(y)$ are continuous for $z \in \mathcal{X}''$.

The weak* topology is weaker than the weak topology, which is weaker than the norm topology. Moreover, the unit ball $\mathcal{B}$ is not compact for the weak topology as soon as the space is not reflexive (otherwise, the weak and weak* topologies coincide).

My questions are the following: Are there intermediate topologies between the weak* and the weak topology for which the unit ball $\mathcal{B}$ is compact? Or can we say in some sense that the weak* topology is the finest for which the unit ball is compact?

I am not expecting a unique answer for every non-reflexive infinite dimensional Banach spaces, but possibly characterisations of the spaces for which the weak* is indeed the only topology between the weak* and the weak topology.

If it helps, the same questions can be considered for the specific cases:

• $(\mathcal{X},\mathcal{X}',\mathcal{X}'') = (c_0(\mathbb{Z}), \ell_1(\mathbb{Z}), \ell_\infty(\mathbb{Z}))$ where $c_0(\mathbb{Z})$ is the space of vanishing sequences endowed with the norm $\|\cdot\|_\infty$.
• $(\mathcal{X},\mathcal{X}',\mathcal{X}'') = (\mathcal{C}(\mathbb{T}), \mathcal{M}(\mathbb{T}), \mathcal{M}'(\mathbb{T}))$ where $\mathbb{T}$ is the torus, $\mathcal{C}(\mathbb{T})$ the space of continuous periodic functional endowed with the supremum norm, and $\mathcal{M}(\mathbb{T}) the space of finite Radon measure.

(Motivation: I try to understand what is the largest topology for which $\mathcal{B}$ is compact beyond the weak* topology in order to use the Krein-Millmann theorem ensuring the existence of extreme points for convex compact sets.)