The book of Douglas says on page 12:
Theorem (Banach): Every Banach space $B$ is isometrically isomorphic to a closed subspace of $C(X)$ for some compact Hausdorff space $X$.
Proof: Let $X$ be $(B^*)_1$ with the w*-topology and define $\beta$: $B \rightarrow C(X)$ by $(\beta f)(\phi) = \phi(f)$ with $\phi \in (B^*)_1$ and $f \in B$.
Next he prove the linearity of $\beta$ and the fact that $\beta$ is an isomorphism using the Hahn-Banach theorem
My question is: What is $\beta$? I don't understand much of its definition. Isn't supposed to be a function from $B \rightarrow C(X)$? Here it seems to me a function $B \rightarrow X$ because it's equal to $\phi(f)$ that is a function in $X = (B^*)_1$.