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Bill Johnson
Bill Johnson
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If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples whichthat I somewhat undertandunderstand ($\ell_1$, $c_0$) this does not happen when $\lambda$ is an extreme point of the closed unit ball of $E^*$.

Question: does there exist a Banach space $E$ and $\lambda \in E^*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?

If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples which I somewhat undertand ($\ell_1$, $c_0$) this does not happen when $\lambda$ is an extreme point of the closed unit ball of $E^*$.

Question: does there exist a Banach space $E$ and $\lambda \in E^*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?

If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples that I somewhat understand ($\ell_1$, $c_0$) this does not happen when $\lambda$ is an extreme point of the closed unit ball of $E^*$.

Question: does there exist a Banach space $E$ and $\lambda \in E^*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?

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Todd Trimble
Todd Trimble
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Extrme Extreme unit linear functional not norrmingnorming a vector

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Itaï BEN YAACOV
Itaï BEN YAACOV
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Extrme unit linear functional not norrming a vector

If $E$ is a non-reflexive Banach space then there exists a linear functional $\lambda \in E^*$ of norm one such that $\lambda v < 1$ for all $v \in E$ of norm one. However, in the only non-reflexive examples which I somewhat undertand ($\ell_1$, $c_0$) this does not happen when $\lambda$ is an extreme point of the closed unit ball of $E^*$.

Question: does there exist a Banach space $E$ and $\lambda \in E^*$ such that $\lambda$ is an extreme point of the unit ball and yet does not norm any (non-zero) vector of $E$?