What's Helly's theorem in the proof of the Goldstine–Weston density theorem

Question

I have a problem in understanding a proof of Goldstine–Weston density theorem. The only thing I don't know in the proof is the part of Helly's theorem to be related. The Goldstine–Weston density theorem is as follows:

Suppose that $X$ is a Banach space and $J_X:X\to X^{\star\star}$ is the canonical embedding. Then $\overline{J_X(X)}^{w^{\star}} = X^{\star\star}$.

Proof:

We need only to prove $~\overline{J_X(U(X))}^{w^{\star}} = U(X^{\star\star})$, where $U(X)=\{ x\in X : \| x \|\leq 1 \}$, $U(X^{\star\star})=\{ x^{\star\star} \in X^{\star\star}: \| x^{\star\star} \|\leq 1 \}$. Let $\Phi \in U(X^{\star\star})$, for any $w^{\star}$ neighbourhood $V$ of $\Phi$ defined by \begin{equation} V=\{ \Psi \in X^{\star\star}: |\left<x^{\star}_{i}, \Psi - \Phi\right>| < \varepsilon, x^{\star}_{i}\in X^{\star}, i=1, \cdots, n \}, \end{equation} and let $r=\max\{ \| x^{\star}_{i} \|: i=1, \cdots, n \}$. $\color{magenta}{\text{By Helly's theorem,}}$ there exists $x_{\varepsilon} \in X$ such that $\| x_{\varepsilon} \| \leq 1+\frac{\varepsilon}{2r}$, $\left<x_{\varepsilon}, x^{\star}_{i}\right> = \left<x^{\star}_{i}, \Phi \right>, i=1, \cdots, n$. Set $x_{0} = \frac{r}{r + \varepsilon/2}x_{\varepsilon}$, then we have $\| x_0 \|\leq 1$ and \begin{equation} |\left< x^{\star}_i, \widehat{x}_0 - \Phi \right>| = |\left<x_0-x_{\varepsilon}, x^{\star}_{i} \right>| \leq \frac{\varepsilon}{2} \| x_0 \| < \varepsilon, \end{equation} where $\widehat{x}_0 \overset{def}{=} J_X(x_0) \in V$. Thus, $\Phi \in \overline{J_X(U(X))}^{w^{\star}}$. That is, $U(X^{\star\star}) \subseteq \overline{J_X(U(X))}^{w^{\star}}$. The reversed inclusion relation is due to Alaoglu's theorem.

My question:

I don't know what's the Helly's theorem mentioned in the proof and also I can't find this theorem in my book anywhere. Thanks in advance.

Edit $1$:

Tomek Kania has given the Helly's theorem which I want to know at first (I still want to know that from which book or place a proof of it can be found ). I have verified that there exists an element $x_{\varepsilon} \in X$. However, I can't prove the inequality $\| x_{\varepsilon} \| \leq 1+\frac{\varepsilon}{2r}$.

Answer 1

This is what is often called Helly's theorem.

Let $X$ be a Banach space. Let also $f_1, \ldots, f_n\in X^*$ and scalars $\alpha_1, \ldots \alpha_n$ be given. Then the following conditions are equivalent:

• there exists $x\in X$ such that $\langle x, f_i \rangle = \alpha_i$ ($i\leqslant n$)
• there exists $\gamma \geqslant 0$ such that for any scalars $\beta_1, \ldots, \beta_n$ we have $$|\sum_{i=1}^n \beta_i \alpha_i| \leqslant \gamma\cdot\|\sum_{i=1}^n \beta_i f_i \|.$$

I think that the fact you need is the following weaker version of the above theorem:

Let $X$ be a Banach space and let $x^{**}\in X^{**}$ be a unit vector. Given a finite set $F\subset X^*$ and $\varepsilon>0$, there exists $x\in X$ such that $\|x\|\leqslant 1+\varepsilon$ and $\langle x,f\rangle = \langle f, x^{**}\rangle$ for all $f\in F$.

Proof. Let $Z = \bigcap_{f\in F}\ker f$. Note that $X/Z$ is finite-dimensional (hence reflexive). Denote by $q\colon X\to X/Z$ the canonical quotient map. Choose now $x\in X$ sitting in $q^{**}(x^{**}) + (X/Z)^{**} = q^{**}(x^{**}) + X/Z$ that has norm not exceeding $1+\varepsilon$. $\square$

Note that the Principle of Local Reflexivity is a powerful extension of this theorem.