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Kacper Kurowski
Kacper Kurowski
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Existence of minimal subset of dual ball such that its closed convex hull in weak-* topology is the whole dual ballintersection of kernels is trivial

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-* topology$\bigcap_{\Lambda \in C} \mathrm{ker}(\Lambda) = \{ 0 \}$.

  Is it always possible to selectfind a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$$\bigcap_{\Lambda \in \widetilde{C}} \mathrm{ker}(\Lambda) = \{ 0 \}$, but, removing any element of $\widetilde{C}$ makes it so that the mentionedthis equality no longer holdsfalse?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.

Existence of minimal subset of dual ball such that its closed convex hull in weak-* topology is the whole dual ball

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-* topology.

  Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.

Existence of minimal subset of dual ball such that the intersection of kernels is trivial

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} \mathrm{ker}(\Lambda) = \{ 0 \}$. Is it always possible to find a subfamily $\widetilde{C} \subseteq C$ such that $\bigcap_{\Lambda \in \widetilde{C}} \mathrm{ker}(\Lambda) = \{ 0 \}$, but removing any element of $\widetilde{C}$ makes this equality false?

edited tags, formatting
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YCor
YCor
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Existence of minimal subset of dual ball such that its closed convex hull in weak$weak-*$* topology is the whole dual ball

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-$*$* topology.

Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.

Existence of minimal subset of dual ball such that its closed convex hull in weak$-*$ topology is the whole dual ball

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-$*$ topology.

Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.

Existence of minimal subset of dual ball such that its closed convex hull in weak-* topology is the whole dual ball

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-* topology.

Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.

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Kacper Kurowski
Kacper Kurowski
  • 820
  • 5
  • 10

Existence of minimal subset of dual ball such that its closed convex hull in weak$-*$ topology is the whole dual ball

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-$*$ topology.

Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.