Weak star closure of the unit sphere in the dual space

Asked Jul 28 '22 at 11:15

Active Jul 28 '22 at 12:30

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Let $X$ be a normed vector space with its dual $X^*$. Let $\mathbb{S}^*$ be the unit sphere of $X^*$. We have known that if $X$ (or $X^*$) is reflexive then the weak-star and weak topology of $X^*$ coincide and thus the weak-star closure (or weak closure) of $\mathbb{S}^*$ is the unit ball. I have the following questions:

If $X$ is reflexive, what is the weak-star sequential closure of $\mathbb{S}^*$?
If $X$ is non-reflexive, what are the weak-star and weak-star sequential closure of $\mathbb{S}^*$?

Thank you in advance.

edited Jul 28 '22 at 12:30

asked Jul 28 '22 at 11:15

Khoa Vu

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See this – Ryszard Szwarc Jul 28 '22 at 11:54
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Thank you so much for your comment! But I am considering the weak-star closure of the dual unit ball, not the weak closure, in the case $X$ is non-reflexive. – Khoa Vu Jul 28 '22 at 12:08
The weak star closure is larger than weak closure, as there are less open sets. – Ryszard Szwarc Jul 28 '22 at 12:16
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Thank you! Please review my edited questions. – Khoa Vu Jul 28 '22 at 12:35
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Looks fine. I do not know much about sequential closure, except that it is smaller than the usual closure. If $X$ is separable then (I have impression) the weak star topology on $B^*$ is metrizable. So sequential weak star closure and weak star closure coincide for bounded sets. – Ryszard Szwarc Jul 28 '22 at 12:57

Weak star closure of the unit sphere in the dual space

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