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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?

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  • $\begingroup$ That IS the usual version of Goldstine's theorem. For a real generalizaton, Google "principle of local reflexivity". $\endgroup$ Jun 2, 2012 at 21:32
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    $\begingroup$ What I've meant by stronger is that the proof gives us more than "the image of closed unit ball under canonical embedding is *-weakly dense in the closed ball of the bidual" which seems to be usual formulation in books. $\endgroup$
    – robibok
    Jun 2, 2012 at 21:44
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    $\begingroup$ I have seen this called "Helly's theorem" while Goldstine's theorem is what is stated in Wikipedia. $\endgroup$ Jun 3, 2012 at 3:11
  • $\begingroup$ Thanks a lot, it's a simple corollary from Helly's theorem. $\endgroup$
    – robibok
    Jun 3, 2012 at 12:20

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