A functional analytic proof of a Borel selection theorem
The author has given a Borel selection theorem, which, together with its immediate consequences, should suffice for most needs in functional analysis. As noted there by him, neither this selection theorem nor its consequences are new, but the argument employed in his proof is different and interesting in the meaning that it only uses a little general topology and some elementary functional analysis. That is, the base of the proof for his main general selection theorem is the following fundamental and simple lemma: Let X be a normed linear space, Y a closed subspace of X having codimension 1, and R the restriction map of \(X^*\) onto \(Y^*\). Let K be a compact metrizable (in the \(weak^*\) topology) subset of \(X^*\) and set \(L=R(K)\). Then there exists a map S from L into K such that \(R(S(\phi))=\phi\) for all \(\phi\in L\) and S is a Borel isomorphism of L onto S(L). In final section, making use of his general selection theorem, many of the standard selection theorems needed in unitary representation theory have been derived.
scientific article; zbMATH DE number 4181682