Goldstine theorem

In functional analysis, a branch of mathematics, the Goldstine theorem, named after Herman Goldstine, asserts that the image of the closed unit ball $B_X$ of a Banach space $X$ under the canonical imbedding into the closed unit ball $B_{X^{**}}$ of the bidual space $X^{**}$ is weakly*-dense.

Proof

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 $x \in (1%2B\delta) B_{X}$ such that $\phi_i(x)=x^{**}(\phi_i)$ for every $i=1,\dots,n$ .

If the requirement $||x|| \leq 1%2B\delta$ 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 $Y�:= \cap_{i} \ker \phi_i = \mathrm{ker} \Phi$ . Every element of $(x%2BY) \cap (1%2B\delta) B_{X}$ has the required property, so that it suffices to show that the latter set is not empty.

Assume that it is empty. Then $\mathrm{dist}(x,Y) \geq 1%2B\delta$ and by the Hahn-Banach theorem there exists a linear form $\phi \in X^*$ such that $\phi|_{Y}=0$ , $\phi(x) \geq 1%2B\delta$ and $||\phi||_{X^*}=1$ . Then $\phi \in \mathrm{span}(\phi_1, \dots, \phi_n)$ and therefore

$1%2B\delta \leq \phi(x) = x^{**}(\phi) \leq ||\phi||_{X^*} ||x^{**}||_{X^{**}} \leq 1,$

which is a contradiction.

Goldstine theorem

Proof

See also