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In [[Functional analysis|functional analysis]], a branch of mathematics, the '''Goldstine theorem''', named after [[Biography:Herman Goldstine|Herman Goldstine]], is stated as follows:
:'''Goldstine theorem.''' Let <math>X</math> be a [[Banach space]], then the image of the closed unit ball <math>B \subseteq X</math> under the canonical embedding into the closed unit ball <math>B^{\prime\prime}</math> of the [[Dual space|bidual space]] <math>X^{\prime\prime}</math> is a weak*-[[Dense set|dense subset]].

The conclusion of the theorem is not true for the norm topology, which can be seen by considering the Banach space  of real sequences that converge to zero, c0 space <math>c_0,</math> and its bi-dual space [[Lp space]] <math>\ell^{\infty}.</math> 

== Proof ==

=== Lemma ===

For all <math>x^{\prime\prime} \in B^{\prime\prime},</math> <math>\varphi_1, \ldots, \varphi_n \in X^{\prime}</math> and <math>\delta > 0,</math> there exists an <math>x \in (1+\delta)B</math> such that <math>\varphi_i(x) = x^{\prime\prime}(\varphi_i)</math> for all <math>1 \leq i \leq n.</math>

==== Proof of lemma ====

By the surjectivity of
<math display="block">\begin{cases}
\Phi : X \to \Complex^{n}, \\ x \mapsto \left(\varphi_1(x), \cdots, \varphi_n(x) \right)
\end{cases}</math>
it is possible to find <math>x \in X</math> with <math>\varphi_i(x) = x^{\prime\prime}(\varphi_i)</math> for <math>1 \leq i \leq n.</math>

Now let
<math display="block">Y := \bigcap_i \ker \varphi_i = \ker \Phi.</math>

Every element of <math>z \in (x + Y) \cap (1 + \delta)B</math> satisfies <math>z \in (1+\delta)B</math> and <math>\varphi_i(z) = \varphi_i(x)= x^{\prime\prime}(\varphi_i),</math> so it suffices to show that the intersection is nonempty.

Assume for contradiction that it is empty. Then <math>\operatorname{dist}(x, Y) \geq 1 + \delta</math> and by the [[Hahn–Banach theorem]] there exists a linear form <math>\varphi \in X^{\prime}</math> such that <math>\varphi\big\vert_Y = 0, \varphi(x) \geq 1 + \delta</math> and <math>\|\varphi\|_{X^{\prime}} = 1.</math> Then <math>\varphi \in \operatorname{span} \left\{ \varphi_1, \ldots, \varphi_n \right\}</math><ref>{{cite book|last1=Rudin|first1=Walter |title=Functional Analysis|location=Lemma 3.9|pages=63–64|edition=Second}}</ref> and therefore
<math display="block">1+\delta \leq \varphi(x) = x^{\prime\prime}(\varphi) \leq \|\varphi\|_{X^{\prime}} \left\|x^{\prime\prime}\right\|_{X^{\prime\prime}} \leq 1,</math>
which is a contradiction.

=== Proof of theorem ===

Fix <math>x^{\prime\prime} \in B^{\prime\prime},</math> <math>\varphi_1, \ldots, \varphi_n \in X^{\prime}</math> and <math>\epsilon > 0.</math> Examine the set
<math display="block">U := \left\{ y^{\prime\prime} \in X^{\prime\prime} : |(x^{\prime\prime} - y^{\prime\prime})(\varphi_i)| < \epsilon, 1 \leq i \leq n \right\}.</math>

Let <math>J : X \rightarrow X^{\prime\prime}</math> be the embedding defined by <math>J(x) = \text{Ev}_x,</math> where <math>\text{Ev}_x(\varphi) = \varphi(x)</math> is the evaluation at <math>x</math> map. Sets of the form <math>U</math> form a base for the weak* topology,<ref>{{cite book|last1=Rudin|first1=Walter|title=Functional Analysis|location=Equation (3) and the remark after|page=69|edition=Second}}</ref> so density follows once it is shown <math>J(B) \cap U \neq \varnothing</math> for all such <math>U.</math> The lemma above says that for any <math>\delta > 0</math> there exists a <math>x \in (1+\delta)B</math> such that <math>x^{\prime\prime}(\varphi_i)=\varphi_i(x),</math> <math>1\leq i\leq n,</math> and in particular <math>\text{Ev}_x \in U.</math>  Since <math>J(B) \subset B^{\prime\prime},</math> we have <math>\text{Ev}_x \in (1+\delta)J(B) \cap U.</math> We can scale to get <math>\frac{1}{1+\delta} \text{Ev}_x \in J(B).</math> The goal is to show that for a sufficiently small <math>\delta > 0,</math> we have <math>\frac{1}{1+\delta} \text{Ev}_x \in J(B) \cap U.</math>

Directly checking, one has
<math display="block">\left|\left[x^{\prime\prime} - \frac{1}{1+\delta} \text{Ev}_x\right](\varphi_i)\right| = \left|\varphi_i(x) - \frac{1}{1+\delta}\varphi_i(x)\right| = \frac{\delta}{1+\delta} |\varphi_i(x)|.</math>

Note that one can choose <math>M</math> sufficiently large so that <math>\|\varphi_i\|_{X^{\prime}} \leq M</math> for <math>1 \leq i \leq n.</math><ref>{{cite book|last1=Folland|first1=Gerald|title=Real Analysis: Modern Techniques and Their Applications|location=Proposition 5.2|pages=153–154|edition=Second}}</ref> Note as well that <math>\|x\|_{X} \leq (1+\delta).</math> If one chooses <math>\delta</math> so that <math>\delta M < \epsilon,</math> then
<math display="block">\frac{\delta}{1+\delta} \left|\varphi_i(x)\right| \leq \frac{\delta}{1+\delta} \|\varphi_i\|_{X^{\prime}} \|x\|_{X} \leq  \delta \|\varphi_i\|_{X^{\prime}} \leq \delta M < \epsilon.</math>

Hence one gets <math>\frac{1}{1+\delta} \text{Ev}_x \in J(B) \cap U</math> as desired.

== See also ==

* {{annotated link|Banach–Alaoglu theorem}}
* {{annotated link|Bishop–Phelps theorem}}
* {{annotated link|Eberlein–Šmulian theorem}}
* {{annotated link|Mazur's lemma}}

== References ==

{{reflist}}

* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->

{{Functional Analysis}}

[[Category:Banach spaces]]
[[Category:Theorems in functional analysis]]

de:Schwach-*-Topologie#Eigenschaften

{{Sourceattribution|Goldstine theorem}}