Open-circuit time constant method

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.

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, and its bi-dual space ℓ_∞.

Proof

Given an $x^{**}\in B_{X^{**}}$, a tuple $({\varphi }_1,\dots ,{\varphi }_n)$ of linearly independent elements of $X^{*}$ and a $\delta >0$ we shall find an $x\in (1+\delta )B_X$ such that ${\varphi }_i(x)=x^{**}({\varphi }_i)$ for every $i=1,\dots ,n$.

If the requirement $\Vert x\Vert \le 1+\delta$ is dropped, the existence of such an $x$ follows from the surjectivity of

$\mathrm{\Phi }:X\to {\mathbb{ℂ}}^n,x\mapsto ({\varphi }_1(x),\dots ,{\varphi }_n(x)).$

Let now $Y:=\bigcap _i\ker {\varphi }_i=\ker \mathrm{\Phi }$. 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 $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,Y)\ge 1+\delta$ and by the Hahn-Banach theorem there exists a linear form $\varphi \in X^{*}$ such that $\varphi {|}_Y=0$, $\varphi (x)\ge 1+\delta$ and $\Vert \varphi {\Vert }_{X^{*}}=1$. Then $\varphi \in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}({\varphi }_1,\dots ,{\varphi }_n)$ and therefore

$1+\delta \le \varphi (x)=x^{**}(\varphi )\le \Vert \varphi {\Vert }_{X^{*}}\Vert x^{**}{\Vert }_{X^{**}}\le 1,$

which is a contradiction.

See also

• Banach–Alaoglu theorem
• Bishop–Phelps theorem
• Eberlein–Šmulian theorem
• Mazur's lemma
• James' theorem