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Define $S_X=\{x\in X:\|x\|=1\}$ and $B_X=\{x\in X:\|x\|\leq 1\}$. A functional $f\in S_{X^*}$ is called $\bf{extreme}$ if it cannot be written as a convex combination of more than or equal to two points in $B_{X^*}$.

I want to prove or disprove that for a Banach space $X$, an extreme functional $f\in X^*$ attains its norm. $X^*$ is the dual space of $X$.

We know that $X$ is reflexive if and only if all $f\in X^*$ attain its norm. But in general all functionals do not attain its norm.

Is there any example of a Banach space where all extreme functionals do not attain its norm?

I am not able to get an example. Please help me. Any help is appreciated. Thank you in advance.

Tuh
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    The only lead that I have would be that $B_{X’}$ is compact for the weak-$*$ topology, but I’m not sure of how it would be relevant – julio_es_sui_glace Nov 27 '24 at 08:13
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    Even though this information is known to me but I could not able to build a counterexample. – Tuh Nov 27 '24 at 08:15
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    I think you can probably find a counterexample with $C([0,1])$ with uniform norm. Using Riesz theorem or something like that. – julio_es_sui_glace Nov 27 '24 at 08:29
  • One thing which I know about $C([0, 1])^$ is that the extreme points of $C([0, 1])^$ with respect to probability measure are the point measure with unit mass. But this point measures attain their norm. Can you please help me further? – Tuh Nov 27 '24 at 08:35
  • I do not have much idea about $C[0, 1]^*$. If possible could you please elaborate the answer? Please – Tuh Nov 27 '24 at 08:53
  • It was also a false lead form me sorry. I went into the details and it does not really work as I intended – julio_es_sui_glace Nov 27 '24 at 08:58
  • It's ok. Totally helpless I am. – Tuh Nov 27 '24 at 09:05

1 Answers1

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This is false. Let $X$ be a non-reflexive Banach space such that $X^*$ is strictly convex. Then every $f \in B_{X^*}$ is an extreme point due to strict convexity but by Jame's theorem there is $f \in B_{X^*}$ that is not norm attaining.

I can't think of an easy example of a Banach space $X$ such that $X^*$ is strictly convex and not reflexive, but such spaces exist. For example, any separable dual space $X^*$ admits a strictly convex renorming.

Evangelopoulos Foivos
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