Let me quote the proof of Goldstine theorem from wikipedia ( http://en.wikipedia.org/wiki/Goldstine_theorem ):
Given an $x^{**} \in B_{X^{**}}$, a tuple $(\phi_1, \dots, \phi_n)$ of linearly independent elements of $X^*$ and a $\delta>0$ we shall find an $x \in (1+\delta) B_{X}$ such that $\phi_i(x)=x^{**}(\phi_i)$ for every $i=1,\dots,n$. If the requirement $||x|| \leq 1+\delta$ is dropped, the existence of such an $x$ follows from the surjectivity of $\Phi : X \to \mathbb{C}^{n}, x \mapsto (\phi_1(x), \dots, \phi_n(x)).$ Let now $Y = \cap_{i} \ker \phi_i = \ker \Phi$. Every element of $x+Y \cap (1+\delta) B_{X}$ has the required property, so that it suffices to show that the latter set is not empty. Assume that it is empty. Then $\mathrm{dist}(x,Y) \geq 1+\delta$ and by the Hahn-Banach theorem there exists a linear form $\phi \in X^*$ such that $\phi|_{Y}=0$, $\phi(x) \geq 1+\delta$ and $||\phi||_{X^*}=1$. Then $\phi \in \mathrm{span}(\phi_1, \dots, \phi_n)$ and therefore $$1+\delta \leq \phi(x) = x^{**}(\phi) \leq ||\phi||_{X^*} ||x^{**}||_{X^{**}} \leq 1,$$
As you can see, it's slightly stronger version than the usual Goldstine theorem and it seems to me, that it can't be deduced from the weaker (classical) version. Does anyone know a book or an article which presents Goldstine theorem in that way?
