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?

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?

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 is true or not, 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?

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?