Direct limits which are Hilbert spaces (Q1813301)

For a metric space \((X,\rho)\) let \(2^ X\) denote the hyperspace of all nonempty compact subsets of \(X\) equipped with the Hausdorff metric. We denote \(\exp(X)=2^ X\) and \(\exp^ n(X)=\exp(\exp^{n-1}(X))\), \(X'=\lim_{n\to\infty}\exp^ n(X)\), \(X^*=\) the completion of \(X'\). In [Proc. Am. Math. Soc. 89, 329-335 (1983; Zbl 0541.54016)] \textit{H. Toruńczyk} and \textit{J. West} proved that if \(X\) is a Peano continuum then \((X^*,X')\cong(\ell_ 2,\ell_ 2^ \sigma)\) where \(\ell_ 2\) denotes the Hilbert space of all square summable sequences of real numbers and \[ \ell_ 2^ \sigma=\left\{ x=(x_ n)\in\ell_ 2:\;\sum_{n=1}^ \infty(nx_ n)^ 2<\infty\right\}. \] Here we write \((X^*,X')\cong(\ell_ 2,\ell_ 2^ \sigma)\) iff there exists a homeomorphism\(f\) from \(X^*\) onto \(\ell_ 2\) such that \(f(X')=\ell_ 2^ \sigma\). In this note we establish similar results for compact convex sets in normed spaces.