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Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space. Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at infinity.

For any $t\in K,$ let $\psi_t$ be an evaluation functional on $C_0(K,E).$

If $X^*$ is a dual space of $X,$ then denote $ext(X^*)$ to be the set of extreme points of the unit ball of $X^*.$

The following corollary is quoted in the book 'Isometries on Banach Spaces: Function Spaces' by Fleming and Jamison, Chapter $2,$ page $33,$

Corollary $2.3.6.$ If $X$ is a subspace of $C_0(K,E),$ then $$ext(X^*) \subset \{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}.$$

I would like to ask whether the reverse inclusion holds, that is, does $$\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$$ hold? If yes, can provide a proof or its reference? Otherwise, can I have counterexample?

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    $\begingroup$ This should depend on the subspace $X\subseteq C_0(K,E)$! $\endgroup$
    – Jochen Wengenroth
    Commented Jan 10, 2018 at 8:22
  • $\begingroup$ @JochenWengenroth: I see. So the reverse inclusion is false in general. May I know what conditions on the subspace $X$ would turn the reverse inclusion true? For example, reflexive? Strictly convex? $\endgroup$
    – Idonknow
    Commented Jan 10, 2018 at 9:50
  • $\begingroup$ What happens in the easier case when $E=\mathbb C$? Then you are comparing $ext(X^*)$ for $X\subseteq C_0(K)$ with the collection of all evaluation functions on $K$. $\endgroup$
    – Matthew Daws
    Commented Jan 10, 2018 at 10:43
  • $\begingroup$ @Matthew Daws perhaps you can write your comment as an answer so that I can give a closure to my question? $\endgroup$
    – Idonknow
    Commented Jun 25, 2020 at 4:16

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